1. Universit`a degli Studi di Roma Tor Vergata
MACROAREA DI SCIENZE MATEMATICHE FISICHE E NATURALI
Corso di Laurea Magistrale in Fisica
Tesi di laurea magistrale
BosonSampling validation with
integrated photonics
Candidato:
Luca Innocenti
Matricola 0206284
Relatore Interno:
Dott. Alessandro Cianchi
Relatore Esterno:
Prof. Fabio Sciarrino
Anno Accademico 2014-2015
7. Introduction
Since the early 1980s, it has been argued [1] that simulating quantum systems is a
very challenging task. One source of difficulty is the number of parameters needed to
characterize a generic quantum system, which grows exponentially with the size of
the system. This means that even only storing the state of a large quantum system is
not feasible with classical computer memories. Furthermore, the number of operations
needed to simulate the temporal evolution of such a system also scales exponentially
with the size. Thus, the only way to avoid this exponential overhead in the evolution
is the use of approximation methods (such as Monte Carlo methods). However, for
many problems of interest, no good approximation scheme are available. It is then still
an open problem whether they can be efficiently simulated with a classical approach.
Hence, it is widely accepted that classical systems cannot in general efficiently simulate
quantum systems. While it is not yet possible to prove it, neither mathematically nor
experimentally, there are strong evidences to believe that this is the case.
This distinction between classical and quantum world has many implications. One
of the most notable examples concerns the possibility that computers exploiting the
weirdnesses of quantum mechanics may be able to carry out computations impossible
with only classical resources. With the current available technologies, the experimental
observation of this quantum advantage (sometimes referred to as quantum supremacy [2,
3]) has proven itself to be rather difficult to achieve. In particular, to observe a post-
classical computation with a universal quantum computer one first needs to solve the
problem of fault-tolerant quantum computation [4], which is known to be possible
in principle [5, 6, 7], but might require decoherence rates that are several orders of
magnitude below what achievable today. In the case of linear optics, a number of no-go
theorems led to the widespread belief that linear interferometry alone could not provide
a path to universal quantum computation. For this reason the result of Aaronson and
Arkhipov (AA), that passive linear optical interferometers with many-photon inputs
cannot be efficiently simulated by a classical computer [8], represented a significant
advance. The related computational problem, that is, sampling from the output proba-
bility distribution of such an apparatus, was named by AA the BosonSampling problem.
A quantum device able to efficiently solve it is referred to as a boson sampler.
More in detail, the BosonSampling computational problem consists in sampling
7
8. from the output probability distribution resulting from the time-evolution of n indis-
tinguishable photons into a random m × m unitary transformation. The hardness of
BosonSampling arises from the fact that the scattering amplitude between an input and
an output state configuration is proportional to the permanent of a suitable n×n matrix,
where the permanent is a particular function, defined similarly to the determinant,
which in the general case is known to be hard to compute classically. This immediately
suggests an experimental scheme to build a boson sampler using only linear optical
elements: just inject n indistinguishable photons into an appropriate linear optical
interferometer, and use photon-counting detectors to detect the resulting output states.
AA showed that, already with 20 < n < 30 and m n, this would provide direct
evidence that a quantum computer can solve a problem faster than what is possible with
any classical device. While this regime is far from our current technological capabilities,
several implementations of 2- and 3-photon devices have soon been reported [9, 10, 11,
12], and other more complex implementations followed [13, 14, 15, 16, 17, 18].
However, the originally proposed scheme to implement BosonSampling, that is,
to generate the input n-photon state through Spontaneous parametric downconver-
sion (SPDC), suffers from scalability problems. Indeed, it is unfeasible to generate high
numbers of indistinguishable input photons with this method, due to the generation
probability decreasing exponentially with n. For this reason, an alternative scheme,
named scattershot boson sampling [19], has been devised [20], and subsequently im-
plemented [18]. Contrarily to a classical boson sampler, a scattershot boson sampler
uses m SPDC sources, one per input mode of the interferometer, to generate random
(but known) n-photon input states, with n m. Each SPDC source generates a pair
of photons, one of which is injected into the interferometer, while the other is used
to herald the SPDC generation event. The use of a scattershot boson sampling scheme
results in an exponential increase of the probability of generating n indistinguishable
photons, for m and n large enough.
While the key part of BosonSampling resides in its simulation complexity, this very
hardness also poses a problem of certification of such a device. Indeed, it is believed [8]
that, when n is large enough, a classical computer cannot even verify that the device is
solving BosonSampling correctly. However, it is still possible to obtain circumstantial
evidence of the correct functioning of a device, and efficiently distinguish the output of
a boson sampler from that resulting from alternative probability distributions, like the
output produced by classical particles evolving through the same interferometer.
A number of validation schemes were subsequently devised to validate the output
resulting from true many-boson interference [14, 16, 21, 22, 23]. In particular, the tests
currently more suitable to identify true many-body interference [22] are those based
on Zero-Transmission Laws (ZTLs) [24]. A ZTL, also often referred to as suppression
law, is a rule which, for certain particular unitary evolutions, is able to predict that
the probability of certain input-output configurations is exactly zero, without having to
8
9. compute any permanent.
However, for the current validation schemes based on ZTLs, it is mandatory that
the input states possess particular symmetries. This requirement may thus be an issue
when the ZTLs are applied to validate a scattershot boson sampler. Indeed, the input
state in scattershot boson sampling is not fixed, but changes randomly at each n-photon
generation event. This mandates for a new ZTL-based validation scheme to be devised,
able to efficiently validate a scattershot boson sampling experiment, but still keeping
the capability of distinguishing alternative probability distributions.
In this thesis we report on both theoretical and experimental advances in the context
of validating classical and scattershot boson sampling experiments:
• From the theoretical point of view, we devise a new validation scheme, more
suitable to validate scattershot boson sampling experiments. This scheme, based
on a ZTL valid for a particular class of matrices, the so-called Sylvester matrices,
generalizes the ZTL reported in [23], presenting significantly higher predictive
capabilities.
• From the experimental point of view, we report on the experimental implementa-
tion [25] of the validation scheme proposed in [22] based on the ZTL for Fourier
matrices [24]. To this end, a scalable methodology to implement the Fourier
transformation on integrated photonics was adopted. This approach exploits the
3-D capabilities of femtosecond laser writing technique, together with a recently
proposed [26] quantum generalization of the Fast Fourier transform algorithm
[27], which allows a significant improvement in the number of elementary optical
elements required to implement the desired Fourier transformation.
The thesis is structured as follows: Chapter 1 opens with a brief survey of classical
and quantum computer science. In chapter 2, after a brief exposition of the theoretical
formalism for many-body quantum states, the fundamental tools used in quantum op-
tics experiments are presented. In chapter 3 the BosonSampling problem is introduced.
The problem of scaling boson sampling experiments is discussed, together with the
recently proposed alternative scheme named scattershot boson sampling. In chapter 4
the subject of boson sampling validation is introduced, and an outline of the proposed
solutions is provided. In particular, the focus is on the validation schemes based on
zero-transmission laws for Fourier and Sylvester matrices, and the possibility of apply-
ing them to scattershot boson sampling experiments. In chapter 5 we present a new
zero-transmission law for Sylvester matrices. Exploiting this zero-transmission law, we
present a scheme to validate scattershot boson sampling experiments. The thesis closes
with chapter 6, where we present the experimental implementation of a validation
scheme for Fourier matrices. The experimental and technological aspects of the experi-
ment are discussed, from the femtosecond laser-written technology employed to build
the integrated interferometers, to a novel method to efficiently implement the Fourier
9
10. transform on an integrated photonics chip. A full analysis of the chip reconstruction
and the observed suppression effects follows. The chapter closes with a discussion of
the usefulness of the presented work, and the possible future improvements.
The work presented in this thesis was carried out at the Quantum Information Lab,
University of Rome La Sapienza.
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11. Chapter 1
Foundations of classical and
quantum information
Quantum information theory is a relatively recent field, and most of the main concepts
have been developed only in the last few decades. It is therefore natural that quantum
information relies heavily on ideas developed in the context of classical information
theory. These provide on one side tools that can be adapted to the quantum realm to
tackle quantum information problems, and on the other a benchmark to measure the
advantages provided by quantum resources with respect to their classical counterparts.
To meaningfully talk about the efficiency of an algorithm in performing a given
task, both in the classical and quantum context, it is necessary to have a mathematically
precise notion of what an “algorithm” is, and a mean to quantify its efficiency in solving
a given computational problem. The formalization of the concept of an algorithm
requires, in turn, the introduction of a computational model. One of the most important
computational models is the so-called Turing machine (TM) model of computation,
discussed in section 1.1. While a variety of other computational models have been
introduced and studied over the years, we will see that, thanks to the Church-Turing
thesis, it is enough to just consider the TM model in defining what algorithms can and
cannot do. Indeed, until a few decades ago, a much stronger conjecture was believed to
be true: the so-called Extended Church-Turing thesis states that any physically plausible
model of computation can be efficiently simulated by a TM. This last conjecture is
however currently under debate, as we now have reasons to believe that computational
devices able to solve certain problems exponentially faster than any classical computer
can indeed be devised. The Church-Turing thesis, as well as its extended version, is
discussed in section 1.2. To meaningfully assess whether an algorithm is more efficient
than another, it is necessary to introduce the idea of complexity classes, studied in the
field of computational complexity theory. These are classes of algorithms sharing some
common scaling properties, and are discussed in section 1.3.
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12. 1.1 Turing machine
While the concept of algorithm as a sequence of operations aimed to obtain a given result
may seem intuitively obvious, a mathematically precise formulation of the concept
was only given in 1930s, thanks to the work of Alonzo Church, Alan Turing, and other
pioneers of the computer era. This work resulted in the development of what may
arguably be considered as the most important computational model of computer science:
the Turing machine (TM) model.
A TM captures the notion of an algorithm performing a computational task, and is
composed of four main elements:
1. a program, which similarly to an ordinary computer program is a sequence of
program lines, each one describing an elementary operation for the TM,
2. a finite state control, which co-ordinates the other operations of the machine,
similarly to a modern computer CPU,
3. a tape, which acts like a computer memory,
4. a read-write tape-head, pointing to the position on the tape which is currently
readable or writable.
The finite state control for a TM consists of a finite set of internal states, q1, . . . , qm. The
number of states m can be varied, however it turns out that for m sufficiently large this
change does not affect the power of the machine in any essential way, so without loss
of generality we may suppose that m is some fixed constant. The finite state control
can be thought of as a sort of microprocessor, co-ordinating the TM’s operation. It
provides temporary storage off-tape, and a central place where all processing for the
machine may be done. In addition to the states q1, . . . , qm, there are also two special
internal states, labelled qs and qh. We call these the starting state and the halting state,
respectively. The idea is that at the beginning of the computation, the TM is in the
starting state qs. The execution of the computation causes the TM’s internal state to
change. If the computation ever finishes, the TM ends up in the state qh to indicate that
the machine has completed its operation.
The TM tape is a one-dimensional object, which stretches off to infinity in one
direction. The tape consists of an infinite sequence of tape squares numbered starting
from 0. Each tape square contains one symbol drawn from an alphabet, Γ, composed of
a finite number of distinct symbols. The read-write tape-head identifies a single square
on the TM tape as the square that is currently being accessed by the machine.
Summarizing, a TM starts its operation with the finite state control in the state
qs, and with the read-write head at the leftmost tape square. The computation then
proceeds step by step according to a predefined program. The computation is halted
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13. when the current state is qh, and the output of the computation is the current contents
of the tape.
A program for a TM is a finite ordered list of program lines of the form q, x, q , x , s ,
where q, q are internal states of the machine, x, x are symbols of the alphabet Γ, and s
is equal to −1, +1, or 0.
At any given step of the computation, if q and x are the current internal state and
the symbol under the read-write tape-head respectively, the TM looks through the list
of program lines in order, searching for a line of the form q, x, ·, ·, · . If it does not find
such a line, the internal state is changed to qh and the machine halts. If such a line is
found, than that program line is executed: the internal state is changed to q , the current
symbol on the tape is overwritten by the symbol x , and the tape-head moves left, right,
or stands still, depending on whether s is −1, +1, or 0, respectively.
Example 1 (Computation of the constant function f(x) = 1). Consider a TM with
three internal states, q1, q2, and q3, in addition to the starting state qs and the halting
state qh. The used alphabet will contain four symbols: Γ = { , b, 0, 1}, with the zeros
and ones used to denote the input number, the to mark the beginning of the input
number, and the blanks (b) used on all the other cells of the tape. Finally, we program
the TM with the following program lines:
1 : qs, , q1, , +1 ,
2 : q1, 0, q1, b, +1 ,
3 : q1, 1, q1, b, +1 ,
4 : q1, b, q2, b, −1 ,
5 : q2, b, q2, b, −1 ,
6 : q2, , q3, , +1 ,
7 : q3, b, qh, 1, 0 .
If we start the TM with the tape containing a number expressed in binary form,
surrounded by blank (b) tapes, with the symbol marking the beginning of the input
number, and the tape-head starting on the cell containing , like the following:
b b b 1 1 1 0 1 b b b b
qs
start
. . .. . .
we will get as output the following state:
13
14. b b b 1 b b b b b b b b
qh
end
. . .. . .
To see this, we have to analyse how the program given to the TM acts on the
initial tape: starting in the state qs on a tape cell with the symbol , the first line of
the program mandates the tape-head to move right and switch to the state q1. In
the q1 state, following the lines 2 and 3, the tape-head will move right over-writing
all the ones and zeros it finds, until it reaches a blank cell. When a blank cell is
reached, according to line 4, the tape-head changes its state to q2 and starts moving
left, continuing moving left, following line 5, until it reaches again the cell. When
the cell is reached, the state is changed to q3 and the tape-head is moved once on the
right. At this point, because of the line 7 of the program, the cell under the tape-head
- that is, the cell on the right of the one with - is over-written to 1, and the tape-head
state changed to qh, halting the execution.
The above analysis shows that this program computes the constant function
f(x) = 1. That is, regardless of what number is given in input onto the tape, the TM
halts with the number 1 represented onto the tape.
In general, a TM can be thought of as computing functions from the non-negative
integers to the non-negative integers, with the input to the function represented by the
initial state of the tape, and the output of the function by the final state of the tape.
The above presented TM is used to formalize the concept of a deterministic algorithm.
To also consider non-deterministic algorithms, this model must however be extended. For
this purpose, the TM model is generalized to that of a probabilistic TM. In a probabilistic
TM, the state transitions are choosen according to some probability distribution, instead
of being completely predetermined.
A further generalization of TMs provides a theoretical basis for quantum algorithms.
These are a special kind of algorithms which, exploiting the properties of quantum
mechanics, can potentially outperform any classical algorithm in certain tasks.
1.2 Church-Turing thesis
An interesting question is what class of functions is it possible to compute using a Turing
machine. Despite its apparent simplicity, the TM model can be used to simulate all
the operations performed on a modern computer. Indeed, according to a thesis put
forward independently by Church and Turing, the TM model completely captures the
notion of computing a function using an algorithm. This is known as the Church-Turing
thesis [28]:
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15. Church-Turing thesis: The class of functions computable by a Turing ma-
chine corresponds exactly to the class of functions which we would naturally
regard as being computable by an algorithm.
The Church-Turing thesis asserts an equivalence between the rigorous mathematical
concept of “function computable by a Turing machine”, and the intuitive concept of
what it means for a function to be computable by an algorithm. In this sense it is
nothing more than a definition of what we mean when we talk of the “computability” of
a function. This thesis is relevant because it makes the study of real-world algorithms
amenable to rigorous mathematical analysis.
We remark that it is not obvious that every function which we would intuitively
regard as computable by an algorithm can be computed using a TM. Indeed, it is conceiv-
able that in the future we will discover in Nature a process which computes a function
not computable by a TM. Up to now, however, no such process has been observed.
Indeed, as will be discussed in more detail in later sections, quantum computers also
obey the Church-Turing thesis. That is, quantum computers can compute the same
class of functions computable by a TM.
A much stronger statement than the Church-Turing thesis is the so-called Extended
Church-Turing thesis (ECT) [8] (also sometimes referred to as Strong Church-Turing
thesis [28]):
Extended Church-Turing thesis: All computational problems that are
efficiently solvable by realistic physical devices, are efficiently solvable by a
Turing machine.
The ECT was however already found to be insufficient to capture all realistic com-
putational models in the 1970s, when Solovay and Strassen [29] devised an efficient,
probabilistic primality test. As the Solovay-Strassen algorithm relied essentially on
randomness, it provided the first evidence that probabilistic Turing machines are capa-
ble to solve certain problems more efficiently than deterministic ones. This led to the
following ad-hoc modification to the ECT:
Extended Probabilistic Church-Turing thesis: All computational prob-
lems that are efficiently solvable by realistic physical devices, are efficiently
solvable by a probabilistic Turing machine.
As this is the form the ECT is currently usually stated as, this is the version we will refer
to when talking in the following of “ECT”.
However, even in this modified form, the ECT still does seem to be in contrast with
the currently accepted physical laws. The first evidence in this direction was given by
Shor [30], which proved that two very important problems - the problem of finding
the prime factors of an integer, and the so-called discrete logarithm problem - could
be solved efficiently on a quantum computer. Since no efficient classical algorithm -
neither deterministic nor probabilistic - is currently known to be able to efficiently solve
15
16. these problems, Shor’s algorithm strongly suggests that quantum mechanics allows
to solve certain problems exponentially faster than any classical computer, and this
directly contradicts the ECT.
1.3 Complexity theory
Computational complexity theory analyzes the time and space resources required to
solve computational problems [28]. Generally speaking, the typical problem faced
in computational complexity theory is proving some lower bounds on the resources
required by the best possible algorithm for solving a problem, even if that algorithm is
not explicitly known.
One difficulty in formulating a theory of computational complexity is that different
computational models may lead to different resource requirements for the same problem.
For instance, multiple-tape TMs can solve many problems significantly faster than single-
tape TMs. On the other hand, the strong Church-Turing thesis states that any model of
computation can be simulated on a probabilistic TM with at most a polynomial increase
in the number of elementary operations required. This means that if we make the
coarse distinction between problems which can be solved using resources which are
bounded by a polynomial in n, and those whose resource requirements grow faster than
any polynomial in n, then this distinction will be well-defined and independent of the
considered computational model. This is the chief distinction made in computational
complexity.
With abuse of the term exponential, the algorithms with resource requirements
growing faster than any polynomial in n are said to require an amount of resources
scaling exponentially in the problem size. This includes function like nlog n
, which grow
faster than any polynomial but lower than a true exponential, and are nonetheless said
to be scaling exponentially, in this context. A problem is regarded as easy, tractable, or
feasible, if an algorithm for solving the problem using polynomial resources exists, and
as hard, intractable, or infeasible, if the best possible algorithm requires exponential
resources.
Many computational problems are formulated as decision problems, that is problems
with a yes or no answer. For example, the question is a given number m a prime number
or not? is a decision problem. Although most decision problems can easily be stated in
simple, familiar language, discussions of the general properties of decision problems are
greatly helped by the terminology of formal languages. In this terminology, a language
L over the alphabet Σ is a subset of the set Σ∗
of all finite strings of symbols from Σ.
For example, if Σ = {0, 1}, then the set of binary representations of even numbers
L = {0, 10, 100, 110, . . . } is a language over Σ. A language L is said to be decided by a
TM if for every possible input x ∈ Σ∗
, the TM is able to decide whether x belongs to
L or not. In other words, the language L is decided if the TM will eventually halt in a
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17. state encoding a “yes” answer if x ∈ L, and eventually halt to a state encoding a “no”
answer otherwise.
Decision problems are naturally encoded as problems about languages. For instance,
the primality decision problem can be encoded using the binary alphabet Σ = {0, 1},
interpreting strings from Σ∗
as non-negative integers, and defining the language L to
consist of all binary strings such that the corresponding number is prime. The primality
decision problem is then translated to the problem of finding a TM which decides the
language L. More generally, to each decision problem is associated a language L over
an alphabet Σ∗
, and the problem is translated to that of finding a TM which decides L.
To study the relations between computational problems, it is useful to classify them
into complexity classes, each one grouping all problems (that is, in the case of decision
problems, all languages) sharing some common properties. Most of computational
complexity theory is aimed at defining various complexity classes, and at understanding
of the relationships between different complexity classes.
A brief description of the most important complexity classes for decision problems
is provided in the following:
• P: We say that a given problem is in TIME(f(n)) if there is a deterministic TM
which decides whether a candidate x is in the corresponding language in time
O(f(n)), with n the length of x. A problem is said to be solvable in polynomial
time if it is in TIME(nk
) for some k. The collection of all languages which are in
TIME(nk
), for some k, is denoted P, which is an example of a complexity class
Some examples of problems in P are linear programming, the calculation of the
greatest common divisors of two numbers, and the problem of determining if a
number is prime or not.
Not surprisingly, there are lots of problems for which no polynomial-time algo-
rithm is known. Proving that a given decision problem is not in P, however, is very
difficult. A couple of examples of such problems are 1) given a non-deterministic
Turing machine M and an integer n written in binary, does M accept the empty
string in at most n steps? and 2) given a pair of regular expressions, do they represent
different sets?. Many other problems are believed to not be in P. Among these
are notable ones such as Factoring, which is the problem of finding the prime
factors decomposition of an integer. This problem is believed to hard problem for
classical computers, though no proof, nor compelling evidences for it, are known
to date. Factoring is particularly important, since its hardness lies at the heart
of wisely used algorithms in cryptography such as the RSA cryptosystem [28].
• NP: An interesting property of the prime factorization problem is that, even if
finding the prime factorization of an integer n is very hard, it is easy to check if a
proposed set of primes is indeed the correct factorization of n: just multiply the
numbers and check if they equal n. The class of decision problems sharing this
17
18. property is called NP. More generally NP, standing for “nondeterministic poly-
nomial time”, is the class of all decision problems for which there are efficiently
verifiable proofs. A NP problem can often be intuitively stated in the form are
there any solutions that satisfy certain constraints?
While it is clear that P is a subset of NP, the converse is currently not known.
Indeed, whether P equals NP is arguably the most famous open problem in
computer science, often abbreviated as the P = NP problem. Many computer
scientists believe [31, 32, 33] that P = NP. However, despite decades of work,
nobody has been able to prove this, and the possibility that P = NP cannot be
excluded. Some implications of either of these possibilities are shown in fig. 1.1.
A related complexity class is NP-hard, which groups all decision problems that
are, informally, at least as hard as the hardest problems in NP. More precisely, a
problem L is NP-hard if every problem in NP can be reduced to L in polynomial
time. As a consequence, a polynomial algorithm solving an NP-hard would
also automatically provide a polynomial algorithm for all problems in NP. While
this is considered highly unlikely, as many NP problems are believed to not be
solvable in polynomial time, it has never been proved that this is not the case.
Finally, the intersection between NP and NP-hard is the class of the so-called
NP-complete problems.
• BPP: If we extend our definition of a TM allowing it to have access to a source of
randomness, let’s say the ability to flip a fair coin, other complexity classes can
be defined. Such a probabilistic Turing machine may only accept or reject inputs
with a certain probability, but if the probability of an incorrect accept or reject is
low enough, they are as useful as their deterministic counterparts. One of the
most important such classes is BPP, which stands for Bounded-error Probabilistic
Polynomial time. BPP is the class of decision problems solvable by a probabilistic
TM in polynomial time with a probability of error less than 1/3.
The choice of 1/3 as error bound is mostly arbitrary, as any error bound strictly
less than 1/2 can be reduced to practically zero with only a small increase in
the resource requirements. For this reason, problems in BPP are regarded as
as efficiently solvable as P problems. In fact, for practical purposes, BPP is
considered, even more than P, as the class of problems which are efficiently
solvable on a classical computer.
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19. Complexity
P ≠ NP P = NP
NP-Hard
NP-Complete
P
NP
NP-Hard
P = NP =
NP-Complete
Figure 1.1: Relations between the fundamental complexity classes.
All the above considered computational classes only took into account classical
Turing machines. The advent of quantum mechanics and the conception of quantum
computers, however, led to the question of what classes of problems can a quantum
computer solve? To try to answer this question, one must study another kind of
complexity classes, entering the realm of quantum complexity theory. In this context,
arguably the most fundamental complexity class if BQP, standing for Bounded-error
Quantum Polynomial time. This is the quantum generalization of BPP, and is defined
as the set of decision problems solvable by a quantum computer in polynomial time,
with an error probability of at most 1/3 for all instances. Probably the most notable
problem which has been shown to be in BQP is Factoring. Indeed, Shor’s algorithm
[30] was one of the first devised quantum algorithms able to efficiently solve a problem
that the best-known classical counterparts can solve only in exponential time.
While only decision problems have been mentioned to this point, these are not the
only kind of computational problems. Function problems are a generalized version of
decision problems, where the output of the algorithm is not bounded to be a simple
YES/NO answer. More formally, a function problem P is defined as a relation R over
the cartesian product over strings of an alphabet Σ, that is R ⊂ Σ∗
× Σ∗
. An algorithm
is said to solve P if for every pair (x, y) ∈ R, it produces y when given x as input.
A class of function problems that will be of interest in the following are the so-called
counting problems, which are problems that can be stated as how many X satisfy a certain
19
20. property P? An example of such a complexity class is #P (pronounced “sharp P”), which
is the set of counting problems associated with the decision problems in NP. Intuitively,
to each NP problem which can be stated in the form “Are there any solutions having
the property P?” is associated a #P problem which can be stated in the form “How many
solutions are there which satisfy the property P?”. As can be intuitively deduced from
this definition, #P problems are generally believed to be even more difficult than NP
problems.
A pair of computational classes related to #P are #P-hard and #P-complete. These
are defined in a completely analogous way to NP-hard and NP-complete, containing
the class of counting problems at least as hard as any #P problem.
A notable instance of a #P-hard problem is the calculation of the permanent of a
complex-valued matrix. The permanent is a function of matrices defined similarly to
the determinant (see the discussions in the following sections, like definition 2), but
which, contrarily to the latter, is known to not be computable in polynomial time, for
general matrices [34]. Indeed, the problem of computing the permanent of a matrix is
known to be a #P-hard problem, and even #P-complete in special circumstances [34].
1.4 Quantum information and quantum computation
Quantum information theory [28, 35, 36, 37] is the study of the information processing
tasks that can be accomplished using quantum mechanical systems. One of the main
goals of quantum information theory is to investigate how information is stored in the
state of a quantum system, how does it differ from that stored in a classical system, and
how can this difference be exploited to build quantum devices with capabilities superior
to that of their classical counterparts. To this end several concepts and ideas are drawn
from other disciplines, such as quantum mechanics, computer science, information
theory, and cryptography, and merged with the goal of generalizing the concepts of
information and computing to the quantum realm.
In the last few decades, information and computation theory have undergone a
spurt of new growth, expanding to treat the intact transmission and processing of
quantum states, and the interaction of such quantum information with traditional forms
of information. We now know that a fully quantum theory of information offers, among
other benefits, a brand of cryptography whose security rests on fundamental physics,
and a reasonable hope of constructing quantum computers that could dramatically speed-
up the solution of certain mathematical problems. Moreover, at a more fundamental
level, it has become clear that an information theory based on quantum principles
extends and completes classical information theory, much like complex numbers extend
and complete the reals. One of the conceptual building blocks of quantum information
and quantum computation is that of a qubit. This is the quantum generalization of
the classical concept of bit, and the fundamental processing unit of most quantum
20
21. devices. While a bit can be in one of two states, traditionally referred to as 0 and 1, a
qubit is allowed to be in a superposition of these basis states. Properly handling such
qubits, quantum computers are able to process information in ways impossible with
any classical computer.
The first to envisage the notion of a quantum computer was Feynman [1], as a
possible solution to the problem of the exponentially increasing amount of resources
required to simulate complex quantum systems with classical computers. More than a
decade later, Lloyd [38] showed that a quantum computer can indeed act as a universal
quantum simulator, where the word universal refers to the fact that the same machine
is capable of tackling vastly different problems by simply changing the program it runs.
There are a lot of candidate implementations for quantum computation. Among
these, in no particular order, are implementations using superconductors, trapped
ions, quantum dots, nuclear magnetic resonance, diamond nitrogen vacancies, silicon,
linear optics, and many other proposed technologies. Here we will only focus on linear
optical implementations of quantum computing, to highlight the difficulties inherent
to implement universal quantum computers, as opposite to the relatively much easier
demands of boson sampling devices, which will be described in the following sections.
Linear optics quantum computation (LOQC) with single photons has the advan-
tage that photons have very long decoherence times, which means that the quantum
information stored in photons tends to stay there, and that linear optical elements are
arguably the simplest building blocks to realize quantum information processing. The
downside is that photons do not naturally interact with each other, and in order to
apply two-qubit quantum gates, which are necessary to implement universal quantum
computation, such interactions are essential. Because of this, effective interactions
among photons have to be introduced somehow.
The two main methods to implement such interactions among photons are 1) using
Kerr nonlinearities, and 2) the use of projective measurements with photodetectors.
Unfortunately, present-day nonlinear Kerr media exhibit very poor efficiency [28] and
very weak nonlinearities, while projective measurements have the disadvantage of
producing probabilistic quantum gates: more often than not these gates fail, destroying
the quantum information.
In the case of projective-measurements-induced nonlinearities there is however
a way to avoid the issue of nondeterministic gates, still mantaning feasible resource
requirements: the Knill, Laflamme, and Milburn [7] (KLM) scheme. Introduced in 2001,
the KLM protocol allows scalable linear optics quantum computing by using quantum
gate teleportation to increase the probability of success of nondeterministic gates [7, 39,
40]. The downside of the KLM scheme is that, for its implementation, it is still necessary
to overcome a series of experimental challenges, such as the synchronization of pulses,
mode-matching, quickly controllable delay lines, tunable beam splitters and phase
shifters, single-photon sources, accurate, fast, single-photon detectors, and extremely
21
22. fast feedback control of these detectors. While most of these features are not terribly
unrealistic to implement, the experimental state of the art is simply not at the point at
which more complex gate operations such as two-qubit operations can be implemented.
On the other hand, a quantum computer is not necessary to implement quantum
simulation. Dropping the requirement of being able to simulate any kind of system,
special purpose devices can be built to tackle specific problems better than the clas-
sical counterparts, in the simpler conceivable case by just emulating, in an analog
manner, the behaviour of a complex quantum system on a simpler quantum device.
Being generally these special purpose devices easier to implement than full-fledged
quantum computers, it is expected that practical quantum simulation will become a
reality well before quantum computers. However, despite the undeniable practical
usefullness of implementing quantum simulation on a classically intractable quantum
system, this would hardly give a definite answer to the question: are there tasks which
quantum computers can solve exponentially faster than any classical computer? Indeed,
a quantum system that is hard to classically simulate is also typically hard to define as
a computational problem. This makes extremely difficult to definitively prove whether
a classical algorithm, able to efficiently carry out such a simulation, exists.
It is for this reasons that the proposal of a boson computer by Aaronson and Arkhipov
[8] gained much interest in the quantum optics community. This kind of special purpose
linear optics quantum computer requires only to send n indistinguishable photons
through a random unitary evolution, and detect the output photons with standard
photodetectors. No teleportation or feedback mechanisms are required, which makes
the experimental implementation of such a device much easier than that of a quantum
computer following the KLM scheme. Furthermore, the related computational problem
is simple enough to be analytically tractable with the tools of computational complexity,
allowing to obtain very strong theoretical evidence of its hardness.
22
23. Chapter 2
Quantum and non-linear optics
In section 2.1 the formalism required to describe many-body quantum states is intro-
duced. In particular, the focus is on many-boson states, and their evolution through
a unitary transformation. In section 2.2 a derivation of the total number of physical
states according to the bosonic, fermionic, and classical statistics is presented. In sec-
tion 2.3 the focus shifts to a more experimental point of view, and the main tools used
in quantum optics experiments are presented. The chapter closes with an outline of the
Hong-Ou-Mandel effect in section 2.4, which is a striking evidence of how quantum
interference results in highly non-classical consequences.
2.1 Description of many-body states
To describe a quantum mechanical system with a fixed number of particles, it is
enough to use a ket state |Ψ(t) , corresponding to a wave function Ψ(r1, . . . , rn, t) ≡
r1, . . . , rn|Ψ(t) . This is interpreted through Born’s rule as the probability amplitude
of the i-th particle being found at the position ri at the time t, that is, as a function such
that
|Ψ(r1, . . . , rn, t)|2
= probability of finding the i-th particle at the position ri, at the time t.
(2.1)
Such a wave function has to satisfy certain symmetrization rules, depending on the
statistical nature of the particles described, and its time-evolution is characterized by
the Schrödinger equation,
i
∂
∂t
|Ψ(t) = H |Ψ(t) , (2.2)
where H is the Hamiltonian of the system. The Hamiltonian is an operator, correspond-
ing to the total energy of the system in most of the cases, which characterizes the
dynamics of the system. Equation (2.2), together with an initial condition |Ψ(t0) , is
23
24. sufficient to obtain the wave function at each time t: its solution is, at least formally,
given by |Ψ(t) = e−i(t−t0)H/
|Ψ(t0) .
The above described formalism is also called first-quantization, to distinguish it
from another way of dealing with quantum systems, named second-quantization. The
latter differs from the former by a shift in focus: instead of considering the number
of particles as a fixed property of the system and using the wave function to describe
their states, the system is characterized by the number of particles contained in each
possible mode, which are however no longer necessarily fixed. To this end, a creation
operator is defined, for each mode, as an operator which acts on a quantum state and
produces another quantum state differing from the former for a single quantum added
to that mode. The hermitian conjugate of a creation operator is called an annihilation
operator (also destruction operator), and instead of adding a quantum to a given mode,
it does the opposite, producing a new state with one less particle in the mode.
The exact rules followed by creation and annihilation operators depend on the
statistical nature of the particles. For bosons, the creation (annihilation) operator of a
mode labelled i is denoted ˆa†
i (ˆai). The core rules obeyed by these operators are the
following:
ˆai |ni =
√
n |(n − 1)i , ˆa†
i |ni =
√
n + 1 |(n + 1)i ,
[ˆak, ˆa†
q] = δk,q, [ˆak, ˆaq] = [ˆa†
k, ˆa†
q] = 0 ,
(2.3)
where |ni is a state with n particles in the mode labelled i. Quantum states with a
well-defined number of particles in each mode are called Fock states, or number states,
and the set of all Fock states is called Fock space.
The above defined creation operators can be used to denote many-boson states, that
is, quantum states with more than one indistinguishable boson. If any single boson can
be in one of m modes, an n-boson state r having ri particles in the i-th mode will be
written as
|r ≡ |r1, . . . , rm =
1
√
r!
m
k=1
ˆa†
k
rk
|0 . (2.4)
where the notation r! ≡ r1! · · · rk! has been used. Such a list r of m elements, with each
element equal to the number of particles in a given mode, will be referred to as the
Mode Occupation List (MOL) associated to the quantum state. A many-body state such
that for every k = 1, . . . , m, rk = 1 or rk = 0, is said to be a collision-free state.
Another way to characterize many-body states is through a so-called Mode Assign-
ment List (MAL) R . This is a list of n elements, with the i-th element being the mode
occupied by the i-th particle. It is worth noting that for indistinguishable particles one
cannot talk of “the mode of the i-th particle”, however. Because of this, the order of
the elements of MALs cannot have any physical significance. In other words, MALs are
always defined up to the order of the elements, or, equivalently, they must be always
considered conventionally sorted (for example, in increasing order). Representing the
24
25. state with a MAL, eq. (2.4) can be rewritten in the following form:
|r ≡ |R =
1
√
r!
n
k=1
ˆa†
Rk
|0 =
1
µ(R)
n
k=1
ˆa†
Rk
|0 , (2.5)
where we denoted with µ(R) the product of the factorials of the occupation numbers
of the state R, that is, µ(R) ≡ r1! · · · rm!.
Yet another way to describe many-body states that will sometimes be useful is to
explicitly list the occupation numbers of each mode. For example, for a state with three
particles, one in the second mode and two in the fourth mode, we write |12, 24 . If
we want to emphasize the absence of particles in, say, the third mode, we write it as
|12, 03, 24 .
Definition 1 (MOL and MAL representations). All of the many-particle quantum
states used in the following will be assumed to have a fixed number of particles n,
with each particle potentially occupying one of m possible modes. Two ways to
represent a state are:
• As the Mode Occupation List (MOL) r ≡ (r1, . . . , rm), i.e. as the m-dimensional
vector whose element rk is the number of particles in the k-th mode. It follows
from this definition that m
k=1 rk = n. We will refer to this representation as
the MOL representation, and denote with Fn,m the set of all MOLs of n photons
into m modes, and with FCF
n,m the set of collision-free MOLs of n photons into
m modes:
Fn,m ≡ (r1, . . . , rm) | ∀i = 1, . . . , m, ri ≥ 0 and
m
i=1
ri = n , (2.6)
FCF
n,m ≡ {(r1, . . . , rm) ∈ Fn,m | ∀i = 1, . . . , m, ri ∈ {0, 1}} . (2.7)
• As the Mode Assignment List (MAL) R ≡ (R1, . . . , Rn), i.e. as the n-dimensional
vector listing the modes occupied by the particles. Given that for indistinguish-
able particles it is not meaningful to assign a specific mode to a specific particle,
the order of the elements of a MAL are conventionally taken to be in increas-
ing order, so to have a one-to-one correspondence between physical states
and MALs. We will refer to this representation as the MAL representation of a
many-particle quantum state and, following the notation of [41], denote with
Gn,m and Qn,m the set of all MALs of n photons into m modes and the set of
collision-free MALs of n photons into m modes, respectively. Equivalently, Gn,m
25
26. and Qn,m can be defined as particular sets of sequences of n integers, that is,
Gn,m ≡ {(R1, . . . , Rn) | 1 ≤ R1 ≤ · · · ≤ Rn ≤ m} ,
Qn,m ≡ {(R1, . . . , Rn) | 1 ≤ R1 < · · · < Rn ≤ m} .
(2.8)
Clearly, there is a one-to-one relation between Fn,m and Gn,m, and between FCF
n,m and
Qn,m, as these are just different ways to denote the same things:
Fn,m Gn,m, FCF
n,m Qn,m. (2.9)
Another class of sequences that will be useful in the following is Γn,m, that is, the
set of all mn
sequences ω = (ω1, . . . , ωn) of integers such that 1 ≤ ωi ≤ m for each
i = 1, . . . , n. We note that the sets Gn,m and Qn,m defined above can be thought of as
the subsets of non-decreasing and strictly increasing sequences of Γn,m, respectively.
A unitary evolution ˆU acts on creation operators in the following way:
ˆa†
j → ˆUˆb†
j
ˆU†
=
m
k=1
Ujk
ˆb†
k, (2.10)
where Ujk are the scattering amplitudes from the mode j to the mode k, and ˆb†
j are the
creation operators of the output states. See examples 2 and 3 for two simple applications
of eq. (2.10).
Example 2. As an example of the application of eq. (2.10) consider how the unitary
ˆU acts on a single-boson state |j = ˆa†
j |0 , where |0 is the vacuum state:
|j = ˆa†
j |0
ˆU
−−−−→ ˆUˆb†
j
ˆU†
|0 =
m
k=1
Ujk
ˆb†
k |0 =
m
k=1
Ujk |k out , (2.11)
which shows that eq. (2.10) is compatible with the usual rules of evolution of single-
particle states.
Example 3. A less trivial example is provided considering the evolution of a two-
boson state, with the two bosons initially in the modes i and j. The relation between
the creation operators and the two-bosons states is
if i = j, |i j = ˆa†
i ˆa†
j |0 , if i = j, |i j ≡ |i i =
(ˆa†
i )2
√
2
|0 .
26
27. These equations can be written more coincisely as |i j =
ˆa†
i ˆa†
j
√
µ(i,j)
|0 , where µ(i, j) is
equal to 1 or 2 if i = j and i = j, respectively.
Using this notation, we have
|i j =
ˆa†
i ˆa†
j
µ(i, j)
|0
ˆU
−−−−→
ˆUˆb†
i
ˆU† ˆUˆb†
j
ˆU†
µ(i, j)
|0 =
1
µ(i, j)
m
k=1
m
l=1
UikUjl
ˆb†
k
ˆb†
l |0
=
m
k=1
m
l=1
UikUjl
µ(i, j)µ(k, l)
|k l out .
(2.12)
We consider the four possibilities:
1. If i = j and k0 = l0, we have
out k0 l0|i j =
m
k=1
m
l=1
UikUjl(δk,k0 δl,l0 + δk,l0 δl,k0 ) = Ui,k0 Uj,l0 + Ui,l0 Uj,k0 ,
2. If i = j and k0 = l0,
out k0 l0|i j =
1
√
2
m
k=1
m
l=1
UikUjl(2δk,k0 δl,k0 ) =
√
2 Ui,k0 Uj,k0 ,
3. If i = j and k0 = l0,
out k0 l0|i j =
1
√
2
m
k=1
m
l=1
UikUil(δk,k0 δl,l0 + δk,l0 δl,k0 ) =
√
2 Ui,k0 Ui,l0 ,
4. If i = j and k0 = l0,
out k0 l0|i j =
1
2
m
k=1
m
l=1
UikUil(δk,k0 δl,k0 ) = Ui,k0 Ui,k0 .
All of these four cases are coincisely described by writing, for general values of
i, j, k0, l0,
out k0 l0|i j =
1
µ(i, j)µ(k, l)
perm
Ui,k0 Ui,l0
Uj,k0 Uj,l0
,
where perm(M) is the permanent of the matrix M (see definition 2).
27
28. Using eq. (2.5) into eq. (2.10) gives
|r
ˆU
−−−−→
1
√
r!
n
k=1
m
j=1
URk,j
ˆb†
j
|0 =
1
√
r!
m
j1=1
m
j2=1
· · ·
m
jn=1
n
k=1
URk,jk
ˆb†
jk
|0
=
1
√
r! ω∈Γn,m
n
k=1
URk,ω(k)
ˆb†
ω(k) |0 ,
(2.13)
where Γn,m is the set of all sequences of n positive integers lesser than or equal to m.
To compute the scattering amplitudes A(r → s, U) of going from the input r to the
output s ≡ (s1, . . . , sn), we now have to rearrange the terms on the right hand side of
eq. (2.13). To this end we start from the general combinatorial equation (see [41]):
ω∈Γn,m
f(ω1, . . . , ωn) =
ω∈Gn,m
1
µ(ω) σ∈Sn
f(ωσ(1), . . . , ωσ(n)), (2.14)
where f(ω) ≡ f(ω1, . . . , ωn) is any function of n integer numbers, Gn,m is the sequence
of all non-decreasing sequences of n positive integers lesser than or equal to m, given
in definition 1, and Sn is the symmetric group, that is, the set of permutations of n
elements.
Applying eq. (2.14) to eq. (2.13), with f(ω1, . . . , ωn) = n
k=1 URkωk
ˆb†
ωk
, we obtain
|r
ˆU
−−−−→
1
µ(R) ω∈Gn,m
1
µ(ω) σ∈Sn
n
k=1
URk,ωσ(k)
ˆb†
ωσ(k)
|0
=
ω∈Gn,m
1
µ(R)µ(ω)
σ∈Sn
n
k=1
URk,ωσ(k)
|ω1, . . . , ωn out ,
(2.15)
where in the last step we exploited the commutativity of the product of the creation
operators ˆb†
k.
We thus obtained the following expression for the scattering amplitudes for bosonic
particles:
A(r → s, U) ≡ out s|r ≡ s| ˆU |r =
1
µ(R)µ(S)
σ∈Sn
n
k=1
URk,Sσ(k)
, (2.16)
in which the factor on the right hand side can be recognised as the permanent of an
appropriate matrix built from U.
28
29. Definition 2 (Permanent). The permanent of a square matrix, similarly to the de-
terminant, is a function which associates a number to a matrix. It is defined very
similarly to the determinant, with the exception that all minus signs that are present
for the latter become plus signs in the former.
More precisely, the permanent of a squared n×n matrix A = (aij) is
perm(A) =
σ∈Sn
a1,σ(1) · · · an,σ(n) =
σ∈Sn
n
k=1
ak,σ(k), (2.17)
where Sn is the symmetric group, that is, the set of all permutations of n distinct
objects.
To express eq. (2.16) through the above defined permanent function, it is also useful
to introduce the following notations to refer to particular submatrices built from a given
matrix:
Definition 3. Let Mk,l denote the set of all k×l complex-valued matrices. If k = l
we shall write Mk instead of Mk,k. Now, let A ∈ Mk,l, and let α ∈ Gp,k and β ∈ Gq,l.
Then, we shall denote with A[α|β] the p × q dimensional matrix with elements
A[α|β]i,j ≡ Aαi,βj
. If, moreover, α ∈ Qp,k and β ∈ Qq,l, then A[α|β] is a submatrix
of A. If α = β we will simplify the notation to write A[α] instead of A[α|β].
Again, if α ∈ Qp,k and β ∈ Qq,l, we shall denote with A(α|β) the (k − p)×(l − q)
dimensional submatrix of A complementary to A[α|β], that is, the submatrix obtained
from A by deleting rows α and columns β.
Example 4. Consider the 3×4 dimensional matrix A =
1 2 3 0
4 5 6 i
0 4 2 1
. Then, if
α = (1, 1) ∈ G2,4 and β = (2, 4, 4) ∈ G3,3, we have A[α|β] ≡
2 0 0
2 0 0
. If
instead α = (2, 3) ∈ Q2,4 and β = (1, 2) ∈ Q2,3, we have A[α|β] ≡
4 5
0 4
and
A(α|β) ≡ 3 0 .
Using definitions 2 and 3, we see that for any matrix A ∈ Mm and sequences
α, β ∈ Gn,m, we have
perm(A[α|β]) =
σ∈Sn
n
k=1
Aαk,βσ(k)
,
29
30. which is just the factor in brackets on the right hand side of eq. (2.16). We conclude that
A(r → s, U) ≡ out s|r ≡ s| ˆU |r =
1
µ(R)µ(S)
perm(U[R|S]). (2.18)
It is worth noting that the symmetric nature of the bosonic creation operators - that
is, the fact that [ˆa†
i , ˆa†
j] = δij - was essential in the above derivation. Analogous
reasonings carried out using fermionic particles - whose creation operators satisfy
{ˆc†
i , ˆc†
j} ≡ c†
i c†
j + c†
jc†
i = δij - would lead to the result
Afermions
(r → s, U) = det(U[R|S]). (2.19)
While eqs. (2.18) and (2.19) may seem very similar at first glance, especially given
the similarities in the definitions of permanents and determinants, they are extremely
different when trying to actually compute these scattering amplitudes. Indeed, while
the determinant of an n dimensional matrix can be efficiently computed, the same is
in general not true for permanents. This is exactly what makes the BosonSampling
problem interesting, and will be described in detail in the next sections.
In conclusion, we can now write down the unitary matrix describing the evolution
of the many-body states, that is, the matrix Uα,β(m, n, U), with α, β ∈ Gn,m, whose
elements contain the permanents (or the determinants, in the case of fermions) of the
corresponding matrix given by eq. (2.18):
Uα,β(m, n, U) ≡
1
µ(α)µ(β)
perm(U[α|β]), α, β ∈ Gn,m. (2.20)
The dependence of U on n, m, and U will often be omitted when clear from the context.
Example 5. Consider the 3×3 unitary matrix Ui,j ≡ ui,j, injected with 2-photon
input states. The resulting many-boson scattering matrix is:
U(3, 2, U) =
u2,3u3,2 + u2,2u3,3 u2,3u3,1 + u2,1u3,3 u2,2u3,1 + u2,1u3,2
√
2u2,3u3,3
√
2u2,2u3,2
√
2u2,1u3,1
u1,3u3,2 + u1,2u3,3 u1,3u3,1 + u1,1u3,3 u1,2u3,1 + u1,1u3,2
√
2u1,3u3,3
√
2u1,2u3,2
√
2u1,1u3,1
u1,3u2,2 + u1,2u2,3 u1,3u2,1 + u1,1u2,3 u1,2u2,1 + u1,1u2,2
√
2u1,3u2,3
√
2u1,2u2,2
√
2u1,1u2,1√
2u3,2u3,3
√
2u3,1u3,3
√
2u3,1u3,2 u2
3,3 u2
3,2 u2
3,1√
2u2,2u2,3
√
2u2,1u2,3
√
2u2,1u2,2 u2
2,3 u2
2,2 u2
2,1√
2u1,2u1,3
√
2u1,1u1,3
√
2u1,1u1,2 u2
1,3 u2
1,2 u2
1,1
2.2 Counting many-body states
Another interesting property of many-body states is their number. Unlike classical
states, it is not meaningful to assign a state to the single particles of a quantum many-
body state. A quantum state is instead described only giving the list of modes occupied
30
31. by the particles, without reference to which particle is in which mode. The above
example, for indistinguishable particles, would therefore read like the following: “the
two particles are in the modes 1 and 3”. This has the remarkable consequence of
changing the total number of possible quantum states of n particles into m modes, with
respect to the classical case.
The number of classical states of n particles into m modes is easily computed: each
one of the n particles can be in one of m modes, independently of the state of the
others,so we have
m × m × · · · × m
n
= mn
(2.21)
possible states.
On the other hand, if we were to describe quantum many-body states with this
notation (which amounts to using the MAL representation defined in definition 1), we
would have to take into account that MALs differing only for a permutation of the
elements represent the same quantum state. Moreover, in the case of many-fermion
states, Pauli exclusion principle mandates that no more than one particle can occupy the
same mode, posing an additional constraint to the possible many-particle configurations.
A simple example of the differences between the above described types of many-body
states is given in table 2.1, for the case m = 4, n = 2.
classical states
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3) (3,4)
(4,1) (4,2) (4,3) (4,4)
many-boson states
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3) (3,4)
(4,1) (4,2) (4,3) (4,4)
many-fermion states
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3) (3,4)
(4,1) (4,2) (4,3) (4,4)
Table 2.1: Comparison of classical, many-boson,
and many-fermion states. Many-particle states for
n = 2 particles into m = 4 modes, in MAL notation.
The striked out states represent non-physical states. In
the many-boson case, these are due to the indistinguisha-
bility of MALs differing only for the order of the elements.
In the many-fermion case, the additional restrictions im-
posed by Pauli’s principle must be taken into account.
To count the number of many-boson states of n particles into m modes, we will
make use of the MOL representation. The problem is thus to find the number of different
31
32. sequences of m non-negative integers, with the sum of the integer equal to n. A visual
schematic of this problem is that given in fig. 2.1: each of the m numbers is represented
as the number of marbles (or whatever object) in the corresponding bin. From this
representation it becomes clear that the question can be equivalently stated as: in how
many ways can n objects be distributed among m bins? An easy way to compute this
number is starting from fig. 2.1: if we permute in every possible way all of the marbles
and the inner separators between the bins, we span the whole set of many-boson states.
The number of such permutations is equal to the number of permutations of m − 1 (the
number of inner separators) plus n (the number of marbles) objects, that is, (m−1+n)!.
However, this overcounts the number of many-boson states, having treated as different
configurations with the same occupation numbers but some marbles, or separators,
interchanged. The final number must therefore be normalized over the number of such
“transparent” permutations. We conclude that the number of many-boson states is
|Fn,m| = |Gn,m| =
(m + n − 1)!
(m − 1)!n!
=
m + n − 1
n
. (2.22)
Similarly, the number of many-fermion states, equal to the number of collision-free
many-boson states, is
|FCF
n,m| = |Qn,m| =
m!
n!(m − n)!
=
m
n
. (2.23)
The difference between these numbers increases exponentially with n, as seen in fig. 2.2.
Many-boson states Classical statesa) b)
Figure 2.1: Examples of many-particle states of 4 particles into 6 modes. The disk represent the
particles, and each bin a different mode. (a) In the case of many-boson states, the disks (that is, the
particles) are all identical to each other, and therefore the number of disks in a given bin completely
specifies the state. (b) For classical states, the distinguishability of the particles implies that there can be
more than one state with the same number of particles for each mode.
32
33. Figure 2.2: Number of classical (blue), many-boson (orange) and many-fermion (green) states, for m = 8
and m = 32, plotted against the number of photons, in logarithmic scale.
2.3 Tools for quantum optics experiments
The basic building blocks used to manipulate light in linear optics are beam splitters,
half- and quarter-wave plates, and phase shifters. For quantum optics experiments,
single photon sources and detectors are also required.
In this section we will give a brief description of these tools, and establish the
mathematical conventions used in the following sections.
2.3.1 Single-mode phase shift
This optical component changes the phase of the electromagnetic field in a given mode,
which means that it acts on the creation operator of a given mode k in the following
way:
ˆa†
in
phaseshifter
−→ ˆa†
out = eiφ ˆNin
ˆa†
ine−iφ ˆNin
= eiφ
ˆa†
in. (2.24)
Physically, a phase shifter can be implemented as a slab of transparent material with
an index of refraction that is different from that of free space.
2.3.2 Beam splitter
A beam splitter (BS), which is a central component of many optical experiments, consists
of a semireflective mirror: when light impinges onto the mirror, part of it will be reflected
and part will be transmitted. In a classical description, a BS can be simply characterized
by the relations
E3 = R31E1 + T31E2,
E4 = T41E1 + R42E2,
(2.25)
where E1, E2 are the amplitudes of the incoming electric fields, E3, E4 the amplitudes
of the outcoming ones, and the complex coefficients Rij, Tij are respectively the re-
33
34. flectances and transmittances along a particular path through the BS (while we will only
consider the electric fields here, completely analogour relations can be written for the
corresponding magnetic fields). In matrix notation eq. (2.25) are written as
E3
E4
=
R31 T32
T41 R42
E1
E2
, (2.26)
where the above 2x2 matrix is known as the beam splitter matrix. In the ideal case
of a lossless BS, considerations of energy conservation lead to the following relations
between the coefficients of the BS:
|R31|2
+ |T41|2
= |R42|2
+ |T32|2
= 1 and R31T ∗
32 + T41R∗
42 = 0, (2.27)
from which it follows that |R31| = |R42| ≡ |R| and |T41| = |T32| ≡ |T |. Using these
relations we can rewrite eq. (2.26) in the form:
E3
E4
=
cos θ ie−iφ
sin θ
ieiφ
sin θ cos θ
E1
E2
, (2.28)
where θ and φ parametrize the set of all possible 2x2 unitary matrices. The general
structure of the BS can be further simplified by additional assumptions on the forms of
the reflection and transmission coefficients. In the case of symmetrical BSs, for example,
we can assume the coefficients to satisfy
R31 = R42 ≡ R = |R| exp(iφR) and T32 = T41 ≡ T = |T | exp(iφT ), (2.29)
with |R|2
+ |T |2
= 1 and φR − φT = ±π/2, which translates in the condition φ =
0 in the notation of eq. (2.28). In the case of a 50:50 beam splitter with reflection
and transmission coefficients of equal magnitude we have |R| = |T | = 1/
√
2 and
φR − φT = π/2, which translates in the conditions φ = 0 and θ = π/4 in the notation
of eq. (2.28).
The relations in eq. (2.25) convert into analogous relations between the quantized
field operators:
ˆa†
3 = cos θˆa†
1 + ie−iφ
sin θˆa†
2 and ˆa†
4 = ieiφ
sin θˆa†
1 + cos θˆa†
2, (2.30)
or, in the case of 50:50 BSs, in
ˆa†
3 =
1
√
2
ˆa†
1 + iˆa†
2 and ˆa†
4 =
1
√
2
iˆa†
1 + ˆa†
2 . (2.31)
Inverting eq. (2.31) leads to
ˆa†
1 =
1
√
2
ˆa†
3 − iˆa†
4 and ˆa†
2 =
1
√
2
−iˆa†
3 + ˆa†
4 . (2.32)
34
35. An equivalent way to denote eq. (2.32) is obtained by denoting with ˆa†
1, ˆa†
2 the two input
modes and with ˆb†
1,ˆb†
2 the two output modes, and describing the evolution through the
beam splitter with the same notation used in eq. (2.10):
ˆa†
1 →
1
√
2
ˆb†
1 − iˆb†
2 and ˆa†
2 →
1
√
2
−iˆb†
1 + ˆb†
2 . (2.33)
An important type of BS is the polarizing beam splitter (PBS). This is a device which
distinguishes different polarization states of the incoming light. If the PBS is cut to
separate horizontal and vertical linear polarizations, the transformation of the incoming
modes, ˆa1 and ˆa2, yields the following outcoming modes:
ˆa1,H → ˆa1,H and ˆa1,V → ˆa2,V ,
ˆa2,H → ˆa2,H and ˆa2,V → ˆa1,V ,
(2.34)
that is, it does not change the spatial mode of the horizontally polarized states, but
switches the spatial mode of vertically polarized ones.
E1
E2
E3
E4
Figure 2.3: Schematic repre-
sentation of a lossless beam
splitter, with the notation
used in the text for incom-
ing and outcoming electric
fields.
2.3.3 Wave plates
A waveplate is an optical component which alters the polarization of a light wave
travelling through it. The two most commonly used types of waveplates are half- and
quarter-wave plates. The former rotates the polarization direction of linearly polarized
light, while the latter converts linearly polarized light into circularly polarized light
and vice versa.
35
36. The same mathematical description given above for BSs applies to the evolution of
light through a waveplate. Instead of having two different spatial modes, ˆa†
1 and ˆa†
2, the
two incoming modes have two different polarizations, that we will denote with ˆax and
ˆay. The equations describing the evolution of the creation operators of the polarization
modes of the field are thus
ˆa†
x = cos θˆa†
x + ie−iφ
sin θˆa†
y and ˆa†
y = ieiφ
sin θˆa†
x + cos θˆa†
y, (2.35)
where x and y are the polarization modes of the light after the evolution through the
waveplate. In the notation of eq. (2.35), half-wave plates correspond to φ = π/2, and
quarter-wave plates to φ = 0.
Figure 2.4: Left: Example of an optical beam splitter as commonly seen in quantum optics laboratories.
Right: Example of a phase shifter, as commonly seen in quantum optics laboratories.
36
37. 2.3.4 Single-photon sources
An ideal single-photon source [42, 43, 44] would be one that
1. Is deterministic (or “on demand”), meaning that it can emit a single photon at any
arbitrary time defined by the user,
2. Has a 100% probability of emitting a single photon and a 0% probability of multiple-
photon emission,
3. Subsequently emitted photons are indistinguishable,
4. The repetition rate is arbitrarily fast.
Given however that no real-world photon source satisfies all of these specifications,
the deviations from the ideal characteristics must be considered when designing exper-
iments.
Single-photon sources are broadly classified into deterministic and probabilistic.
Among the implementations of the former are those based on color centers [45, 46, 47],
quantum dots [48, 49, 50], single atoms [51], single ions [52], single molecules [53],
and atomic ensembles [54], all of which can to some degree emit single photons “on
demand”. On the other hand are the probabilistic single-photon sources. These generally
rely on photons created in pairs via parametric downconversion in bulk crystals and
waveguides, and four-wave mixing in optical fibers. While these sources are probabilistic
- and therefore it is not possible to know exactly when a photon has been emitted -
because the photons are created in pairs, one of the emitted photons can be used to
herald the creation of the other.
While the distinction between deterministic and probabilistic sources is clear in the
abstract, this distinction blurs in practice. This due to the unavoidable experimental
errors that make also “theoretically deterministic sources” be probabilistic in practice.
Although many applications, especially those in the field of quantum-information
science, require an on-demand source of single photons, probabilistic single-photon
sources remain a fundamental tool, and are widely used in many quantum optics
experiments.
Spontaneous parametric downconversion (SPDC) is an important process in quan-
tum optics, typically exploited to generate entangled photon pairs, or heralded single
photons. This is achieved using a nonlinear crystal - that is, a medium in which the
dielectric polarization responds nonlinearly to the electric field - which converts the
photons of a pump beam into pairs of photons of lower energy. A simple model of the
interaction Hamiltonian in such a crystal is
HI ∼ χ(2)
ˆapˆa†
sˆa†
i + hermitian conjugate, (2.36)
37
38. where χ(2)
is the second-order nonlinear susceptibility of the nonlinear medium. Here
ˆap is the annihilation operator of the pump beam, and ˆa†
s and ˆa†
i are the creation
operators of the signal and idler beams respectively (the names signal and idler are
there for historical reasons, and carry no special significance throughout this thesis).
In the simplest case, with the signal and idler beams initially in vacuum states, a single
photon from the pump beam is converted into two optical photons, one in the signal
beam and the other in the idler:
|1 p |0 s |0 i
SPDC
−−−→ ˆapˆa†
sˆa†
i |1 p |0 s |0 i = |0 p |1 s |1 i . (2.37)
The conditions of energy and momentum conservation pose a contraint on the signal and
idler generated photons, which will depend on the pump beam through the following
relations (see also fig. 2.5):
ωp = ωs + ωi,
kp = ks + ki.
(2.38)
kp
ks ki
Figure 2.5: Phase-matching
condition
There are two main types of SPDC processes, named type I and type II:
In type I, the signal and idler photons have the same polarization, orthogonal to
that of the pump.
In type II downconversion, the polarizations of signal and idler photons are instead
orthogonal to each other. Because of birefringence effects, the generated photons are
emitted along two cones, one for the ordinary wave and another for the extraordinary
wave. At the intersection of the cones, the two photons detected are in a polarization-
entangled state of the form
1
√
2
(|V s |H i + |H s |V i), (2.39)
where |H and |V denote an horizontally and vertically polarized state respectively. A
schematic representation of this process is shown in fig. 2.6.
More generally, the state produced by an SPDC source does not always contain two
photons, but has the form
∞
p=0
gp
|p1 |p2 , (2.40)
38
39. with |pi = (ˆa†
i )p
/
√
p!, and 0 ≤ g < 1 an appropriate parameter determining the ratio
of generated photons and dependent, among other things, on the strength of the pump
beam. Typically, g 1, so that the probability of generating many photons is low. For
instance, in typical conditions g ∼ 0.1 and the probability of generating a state of the
form |21 |22 is lower of a factor ∼ 102
than the probability of producing a single pair
|11 |12 .
The main advantages of SPDC sources are the high photon indistinguishability, the
collection efficiency, and relatively simple experimental setups. This technique, however,
suffers from two drawbacks. First, since the nonlinear process is nondeterministic, so is
the photon generation, even though it can be heralded. Second, the laser pump power,
and hence the source’s brilliance, has to be kept low to prevent undesired higher-order
terms in the photon generation process.
Figure 2.6: Representation of type II downconversion. The pump beam (red) impinges on the nonlinear
crystal, generating, due to birefringence effects, photons along two cones. On the upper ordinary cone
(orange), the generated photons are horizontally polarized, while on the lower extraordinary cone (green)
the generated photons are vertically aligned. Postselecting on the two intersections of these cones (blue
dots), a pair of polarization-entangled photons is obtained.
2.3.5 Single-photon detectors
Roughly speaking, single-photon detectors are devices which convert single photons
into an electrical signal of some sort [42]. Quantum information science is one of the
field currently driving much of the research toward improved single-photon-detector
technology. For example, many quantum communication protocols rely heavily on
39
40. detector properties such as detection efficiency.
An ideal single-photon detector [42] would require the following characteristics:
1. The detection efficiency - that is, the probability that a photon incident upon the
detector is successfully detected - is 100%,
2. The dark-count rate - that is, the rate of detector output pulses in the absence of
any incident photons - is zero,
3. The dead time - that is, the time after a photon-detection event during which the
detector is incapable of detecting another photon - is zero,
4. The timing jitter - that is, the variation from event to event in the delay between
the input of the optical signal and the output of the electrical signal - is zero.
Additionally, an ideal single-photon detector would also be able to count the number of
photons in an incident pulse. Detectors able to do this are referred to as photon-counting,
or photon-number resolving, detectors. However, non-photon-number-resolving detec-
tors, which can only distinguish between zero photons and more than zero photons,
are the most commonly used. Indeed, while detecting a single photon is a difficult task,
discriminating the number of incident photons is even more difficult. Examples of non-
photon-number-resolving single-photon detector technologies include single-photon
avalanche photodiodes [55], quantum dots [56], superconducting nanowires [57], and
up-conversion detectors [58, 59, 60].
2.4 Hong-Ou-Mandel effect
The differences between bosons and fermions are not only in the different numbers of
microstates. Their statistical behaviour can differ significantly, as well as be significantly
different from the behaviour of distinguishable particles.
Bosons, roughly speaking, tend to occupy the same state more often than classical
particles, or fermions, do. This behaviour, referred to as bosonic bunching, has been
verified in numerous experimental circumstances, including fundamental ones like
Bose-Einstein condensation [61, 62, 63]. In the context of optical experiments, the most
known effects arising from the symmetric nature of the Bose-Einstein statistics is the
Hong-Ou-Mandel (HOM) effect [64].
In the original experiment, two photons are sent simultaneously through the two
input ports of a symmetric beam splitter. Since no interaction between the two photons
takes place, one would expect no correlation between the detection events at the two
output ports. Instead, the photons are always seen either both on the first output mode,
or both on the second output mode.
40
41. + − +
Figure 2.7: Pictorial representation of the suppression of non-bunched events, when two indistinguish-
able bosons evolve through a symmetric beam splitter. Each of the four images represent a possible
evolution of the bosons, with all of them interfering with each other. The two events which would result
in one boson per output port turn out to interfere destructively (note the minus sign), and are suppressed.
This effect is a direct consequence of the quantum interference between the possible
ways two-photon states can evolve. A pictorial representation of this is given in fig. 2.7:
when the photons are injected into two different ports of a symmetric beam splitter,
the scattering amplitudes corresponding to the output photons being in two different
modes interfere destructively.
We can derive this result applying eq. (2.33) to the two-photon input state |11, 12 =
ˆa†
1ˆa†
2 |0 :
|11, 12 = ˆa†
1ˆa†
2 |0 →
1
√
2
ˆb†
1 − iˆb†
2
1
√
2
−iˆb†
1 + ˆb†
2 |0
=
1
2
−i(ˆb†
1)2
+ ˆb†
1
ˆb†
2 − ˆb†
2
ˆb†
1 − i(ˆb†
2)2
=
−i
2
(ˆb†
1)2
+ (ˆb†
2)2
|0 =
−i
√
2
(|21 + |22 ) ,
(2.41)
where in the last steps we used the rules given in eq. (2.3), and in particular the
commutativity of the bosonic creation operators, which implies that ˆb†
1
ˆb†
2 = ˆb†
2
ˆb†
1 (see
also fig. 2.7 for a pictorial representation of how the suppression of non-bunched events
arises). We thus conclude that when two indistinguishable photons enter a symmetric
beam splitter one in each mode, they always come out in the same mode.
This property of photons (or, more generally, of bosons) is highly non-classical, and
is a notable example of how interesting effects can arise when dealing with many-body
quantum states.
In a real world experiment, the two input photons will never be perfectly indistin-
guishable, though. A more careful analysis, taking into account the potentially different
times at which the photons reach the beam splitter, as well as the coherence time of
each photon wave packet, predicts a smooth transition from the classical behaviour to
the antibunching effect described above [65].
In the general case of partially distinguishable particles, the probability PT (s, x) of
detecting an output state s in an HOM experiment with a time delay quantified by x,
becomes an average of the probability PB for bosons and the probability PD assigned
to distinguishable particles, weighted by two factors |c1|2
and |c2|2
which depend on
41
42. the relative time delay:
PT (s, x) = |c1(x)|2
PB(s) + |c2(x)|2
PD(s). (2.42)
In a typical experimental scenario, with the incoming photons having a Gaussian
frequency distribution around the central frequency, PT (s, x) also has a gaussian profile
[65], as shown in a typical case in fig. 2.8.
In the case of fully distinguishable particles, where PT = PD, no interaction
occurs and the output events of the single photons are not correlated. There are
two possible classical configurations of two photons in the output MOL (1, 1), and
one configuration for both (2, 0) and (0, 2). It follows that PD((1, 1)) = 2/4 and
PD((2, 0)) = PD((0, 2)) = 1/4, as shown in fig. 2.8 for x ≈ ±400µm. On the other
hand, when the particles are fully indistinguishable, PT = PB. The probability of the
various outcomes is now given by eq. (2.41). The output (1, 1) is thus suppressed, while
PB((2, 0)) = PB((0, 2)) = 1/2, as shown in fig. 2.8 for x = 0.
Figure 2.8: Transition to indistinguishability in a HOM experiment. Changing the time delay between
the two input photons a dip in the number of measured output coincidences is seen, corresponding
to the time delay (or, equivalently, the path delay) making the photons indistinguishable. The blue
line is the probability of detecting two photons in one of the two output ports, that is, PT ((2, 0), x) or
equivalently PT ((0, 2), x). The red line is the probability PT ((1, 1), x) of detecting the two photons in
the two different output ports. As expected, the red plot shows a peak corresponding to the antibunching
effect arising when the particles are indistinguishable, while the blue plot show the HOM explained above.
42
43. Chapter 3
BosonSampling
In this chapter we discuss various aspects of the BosonSampling computational prob-
lem. In section 3.1 the problem of experimentally assessing quantum supremacy is
discussed, in order to appreciate the importance of BosonSampling in the modern
research context. Section 3.2 follows with the description of what the BosonSampling
computational problem is, and its advantages in obtaining experimental evidences
of quantum supremacy. In section 3.3 some issues related to the scalability of boson
sampling implementations are described. The chapter closes with a description of
scattershot boson sampling in section 3.4, as an alternative architecture to scale boson
sampling implementations to higher numbers of photons.
3.1 Importance of BosonSampling
It is currently believed that many quantum mechanical systems cannot be efficiently
simulated with a classical computer [1]. This implies that a quantum device is, to the
best of our knowledge, able to solve problems de facto beyond the capabilities of classical
computers. Exploiting this quantum advantage requires however an high degree of
control over the quantum system, not yet manageable with state of the art technology.
In particular, a post-classical computation with a universal quantum computer will
require an high degree of control of a large number of qubits, and this implies that an
experimental evidence of quantum supremacy [3] with an universal quantum computer
will likely require many years. A notable example is given by the large gap between the
number of qubits that can currently be coherently controlled (∼10), and the number
of qubits required for a calculation such as prime factorization, on a scale that would
challenge classical computers (∼106
). Consequently, there is considerable interest in
non-universal quantum computers and quantum simulators that, while able to only
solve specific problems, might be significantly easier to be implemented experimentally.
Such devices could give the first experimental demonstration of the power of quantum
43
44. devices over classical computers, and potentially lead to technologically significant
applications.
Moreover, in the context of searching for experimental evidence of quantum supremacy,
the technological difficulties are not the only issue. To show this, we will consider as an
example Shor’s quantum algorithm [30] to efficiently factorize integer numbers. Even if
we were to get past the technological difficulties of implementing Shor’s algorithm with
sufficiently many qubits, it could be easily argued that such an achievement would not
be a conclusive evidence that quantum mechanics allows post-classical computations.
This because we do not have to date a mathematically sound proof that there cannot be a
classical algorithm to efficiently factorize integers. In the language of complexity theory,
this corresponds to the fact that we do not have a proof that Factoring is not in P,
even though this is believed enough to base modern cryptography is based on this
conjecture. More generally, before 2010, there were no instances of problems efficiently
solved by quantum computers, which were proved to not be efficiently solvable with
classical ones.
This changed when, in 2010, Aaronson and Arkhipov (AA) proposed [8] the Boson-
Sampling problem as a way to obtain an easier experimental evidence of quantum
supremacy. BosonSampling is a computational problem that, while hard to solve for
a classical computer, is efficiently solved by a special-purpose quantum device. AA
showed that BosonSampling is naturally implemented using only linear optical ele-
ments, in a photonic platform named a boson sampler. The experimental realization
of a boson sampler, while still challenging with present-day technologies, requires
much less experimental efforts with respect to those required to build a universal
quantum computer. In fact, the AA scheme requires only linear optical elements and
photon-counting detectors, as opposite to, for example, the Knill, Laflamme & Milburn
approach [7, 40] for universal linear optics quantum computing, which requires among
other things an extremely fast feedback control of the detectors.
44
45. Figure 3.1: Galton board: n identical balls are dropped one by one from the upper corner, and are
randomly scattered into the lower slots. The quantum generalization of this classical “rudimentary
computer” leads to the idea of BosonSampling. Credits: Nicolò Spagnolo.
3.2 The BosonSampling computational problem
Consider the following linear optical experiment: the n-photon state |rAA given by
|rAA ≡ |11, . . . , 1n, 0n+1, . . . , 0m ≡ ˆa†
1 . . . ˆa†
n |01, . . . , 0m , (3.1)
is injected into a passive linear optics network, which implements a unitary map on
the creation operators:
ˆa†
k → ˆUˆb†
k
ˆU†
=
m
j=1
Uk,j
ˆb†
j. (3.2)
with U an Haar-random m×m complex unitary matrix. The evolution induced on
|rAA is
|rAA →
s∈Fn,m
A(r → s, U) |s , (3.3)
where the sum is extended over all many-boson states of n particles into m modes, and
the scattering amplitudes A are, as shown in eq. (2.18), proportional to the permanents
of n×n matrices. AA argued that, for m n, the output of such an apparatus cannot
be efficiently predicted by a classical computer, neither exactly nor approximately [8].
This was rigorously proved in the exact case. The problem of approximately sampling
45
46. from the output probability distribution of a boson sampling apparatus depends instead
on a series of conjectures, for which strong supporting evidence was provided [8].
This problem, which amounts to that of being able to sample from the output proba-
bility distribution given in eq. (2.18), is referred to as the BosonSampling computational
problem. The constraint m n is essential for the hardness result, as otherwise
semi-classical methods become efficient [66, 67].
Roughly speaking, a boson sampling apparatus is a “quantum version” of a Galton
board. A Galton board, named after the English scientist Sir Francis Galton, is an
upright board with evenly spaced pegs into its upper half, and a number of evenly-
spaced rectangular slots in the lower half (see fig. 3.1). This setup can be imagined to
be a rudimentary “computer”, where n identical balls are dropped one by one from
the upper corner, and are randomly scattered into the lower slots. In the quantum
mechanical version, the n balls are indistinguishable bosons “dropped” simultaneously,
and each peg a unitary transformation, typically implemented as a set of beam splitters
and phase shifters.
More precisely, BosonSampling consists in producing a fair sample of the output
probability distribution P(s |U, rAA) ≡ |A(r → s, U)|2
, where s is an output state of
the n bosons, and rAA the above mentioned input state. The unitary ˆU and the input
state rAA are the input of the BosonSampling problem, while a number of output states
sampled from the correct probability distribution are its solution (see fig. 3.2).
46
47. Figure 3.2: Conceptual boson sampling apparatus. (a) The input of the BosonSampling problem is
the input many-photon state (in figure the state |0, 0, 1, 1, 0, 1, 0, 0, 0 ), and a suitably chosen unitary U.
The output is a number of outcomes picked according to the bosonic output probability distribution (in
figure, two examples of such states are provided, with MOLs 101000100 and 110000100). Colleting enough
such events allows to reconstruct the probability distribution. This, however, requires an exponentially
increasing (in n) number of events. (b) Injecting an m-mode unitary with n indistinguishable photons,
the output state is a weighted superposition of all possible outcomes. Measuring in which modes the
photons ended up results in the collapsing of this superposition. The probability of finding the photons
in a certain configuration is given by eq. (3.4). Credits: [10].
47
48. Example 6 (Solution of the BosonSampling problem). Let ˆU be a randomly chosen
unitary transformation, described by the 4×4 matrix
U =
−0.60046+0.220549i −0.108966−0.527875i −0.367627+0.212122i 0.0655053 +0.340358i
−0.576174−0.386133i −0.463806+0.524027i 0.0458648 −0.0209767i 0.148456 −0.0679879i
−0.0337116−0.30791i 0.408837 −0.0733373i −0.0106664+0.578256i 0.543236 −0.319264i
−0.0894887−0.0760134i 0.211627 +0.0494216i 0.680731 +0.139364i 0.0591935 +0.672804i
.
The output probability distribution resulting from the injection of the two-photon
input state with MOL r = (1, 1, 0, 0) is
output state probability amplitude probability
(0,0,0,2) 0.046478 + 0.0651595 i 0.00640597
(0,0,1,1) -0.0300108+0.0707215 i 0.00590217
(0,0,2,0) -0.0175525 + 0.0246647 i 0.000916437
(0,1,0,1) -0.260804-0.194492 i 0.105846
(0,1,1,0) 0.0432791 -0.312955 i 0.099814
(0,2,0,0) 0.462674 + 0.265491 i 0.284553
(1,0,0,1) 0.0195337 -0.147833 i 0.0222362
(1,0,1,0) 0.270811 +0.0424448 i 0.0751401
(1,1,0,0) 0.0218767 -0.0707259 i 0.00548074
(2,0,0,0) 0.609711 + 0.148185 i 0.393706
Measuring the exit modes of the two injected photons at each pulse, we obtain
a series of samples from the above probability distribution. An example of 10 such
samples is the sequence
(0,2,0,0)
(0,1,1,0)
(2,0,0,0)
(2,0,0,0)
(0,2,0,0)
(0,1,1,0)
(2,0,0,0)
(0,1,0,1)
(0,2,0,0)
(0,2,0,0)
The above list is exactly what BosonSampling is all about: obtaining a list of “labels”
distributed according to a particular probability distribution.
In other words, the BosonSampling problem is not solved obtaining the above
listed probabilities, but obtaining a number of samples from this distribution. The
number of such samples is not really important here, even producing a single state
48
49. from the correct probability distribution would theoretically be enough to achieve
a post-classical computation, though possibly making it harder to experimentally
verify.
The hardness of the BosonSampling problem can be traced back to the #P-hardness
of computing the permanent of a generic complex-valued matrix. Indeed, as shown in
eq. (2.18), the probability P(r → s, U) of an input r evolving into s, is proportional to
the permanent of the matrix U[R|S] (recalling definition 3):
P(r → s, U) = |A(r → s, U)|2
=
1
µ(R)µ(S)
|perm(U[R|S])|2
. (3.4)
Computing the permanent of a n×n matrix with the fastest known classical algorithms
[41, 68] requires a number of operations of the order O(n2n
). This means that, for
example, computing the permanent of a 30×30 complex matrix, corresponding to a
single scattering amplitude for a 30-photon state, requires a number of operations of the
order of ∼ 1010
. If the average time required by a classical computer to perform a single
operation is of the order ∼ 107
, the computation of one such scattering amplitude
will require ∼ 10 minutes. While still clearly manageable by a classical computer,
this already shows the potential advantages of a boson sampling apparatus: if the
experimental problems related to coherently evolve 30 indistinguishable photons inside
an interferometer were to be solved, this would allow to sample from the probability
distribution given by eq. (3.4) without actually knowing the probabilities itselves.
AA demonstrated that, should boson sampling be classically easy to solve, this would
have very strong and undesied consequences in computational complexity theory, and
therefore it is most probable that boson sampling is not classically easy to solve.
It is worth stressing that the BosonSampling problem is not that of finding the per-
manents in eq. (2.18), but only that of sampling from the related probability distribution.
In fact, not even a boson sampler is able to efficiently compute these scattering probabil-
ities. This is due to the fact that, to reconstruct a probability distribution spanning over
a number M of events, roughly speaking, the number of samples is required to be at
least of the order of M. But, as shown in eqs. (2.22) and (2.23), M scales exponentially
with n, implying that the number of experimental samples required to reconstruct the
probability distribution becomes exceedingly large very soon. In figs. 3.3 and 3.4 is
shown that, if the number of samples is not large enough, the reconstructed probability
distribution is different from the real one. In fact, generally speaking, there are strong
arguments against the possibility to compute the permanents of complex-valued matri-
ces by means of quantum experiments [69], although attempts have been reported in
this direction [70].
49
50. 0
0.025
0.05
0.075
0.1
0.125
0.15
States
Probabilitydistribution
0.0
0.5
1.0
1.5
2.0
States
Numberofsamples
0
5
10
15
States
Numberofsamples
0
50
100
150
States
Numberofsamples
a) b)
c) d)
Figure 3.3: Example of boson sampling from a randomly chosen 8×8 unitary matrix. (a) Exact
output probability distribution for two photons injected in the first two modes of a random 8-mode
interferometer. (b), (c), (d) Output states sampled from the output probability distribution, for a number
of samples equal to 10 (b), 100 (c) and 1000 (d). As seen, with a low number of samples, the histogram
may appear different from the true probability distribution.
0
0.01
0.02
0.03
States
Probabilitydistribution
0.0
0.2
0.4
0.6
0.8
1.0
States
Numberofsamples
0
1
2
3
4
5
States
Numberofsamples
0
5
10
15
20
25
30
States
Numberofsamples
a) b)
c) d)
Figure 3.4: Example of boson sampling from a randomly chosen 8×8 unitary matrix. (a) Exact
output probability distribution for four photons injected in the first four modes of a random 8-mode
interferometer. (b), (c), (d) Output states sampled from the output probability distribution, for a number
of samples equal to 10 (b), 100 (c) and 1000 (d). The number of samples required to reliably recover the
original probability distribution is much higher than in fig. 3.3, due to the higher number of many-boson
states, which are here 8+3
4 = 330 against the 8+1
2 = 36 of fig. 3.3.
50
51. The complexity of BosonSampling makes it an extremely valuable candidate to gain
experimental evidences of the supremacy of quantum devices over classical computers.
Indeed, it presents several advantages in this regard over, for example, Factoring,
which is the paradigmatic problem that would allow quantum computers to perform a
post-classical computation:
1. The BosonSampling problem is even harder than Factoring, being related to
the #P-hard complexity class, and believed to not be in NP.
2. A boson sampler requires significantly less resources to be implemented than a
universal quantum computer. In particular, it does not require adaptive or feed-
forward mechanisms, nor fault-tolerance methods. This relatively simple design
has already prompted a number of small-scale implementations of increasing
complexity [9, 10, 11, 12, 13, 14, 15, 17, 18, 25, 71].
AA suggested [8] that a 400-modes interferometer fed with 20 single photons
is already at the boundary of the simulation powers of present-day classical
computers. While in this regime it would still be possible to carry out a classical
simulation, the quantum device should be able to perform the sampling task faster
than the classical computer.
3. The theoretical evidence of the hardness of BosonSampling is stronger than that
of factoring integers: while in the former case the result only relies on a small
number of conjectures regarding the hardness of some problems [8], in the latter
case there is no compelling evidence for Factoring to not be in P. While known to
be in BQP, Factoring is only believed to be in NP, and strongly believed to not
be in NP-hard.
While the hardness of Factoring is strong enough to build modern cryptography,
it could also happen that a polynomial-time algorithm will be discovered showing
that Factoring is in P as, basically, the sole evidence for its hardness is the fact
that no efficient classical algorithm is yet known.
3.3 Scaling experimental boson sampling implemen-
tations
The hardness of BosonSampling has another potentially important consequence: it
could provide the first experimental evidence against the ECT [8]. This point, however,
is still subject to some debate [67, 72, 73], due to the somewhat informal nature of
the ECT itself. Indeed, the ECT is not a mathematical statement, but a statement about
how the physical world behaves, in a certain asymptotic limit. Because of this it is
51
