APS March meeting 2020 Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary U is compiled into a short-depth gate sequence V. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence V (i.e., the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM's noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

1. Noise Resilience of Variational Quantum
Compiling
Kunal Sharma
GRA, Los Alamos National Lab
Phd Candidate, Louisiana State University
ksharm7@lsu.edu
APS March Meeting 2020
Joint work with
Sumeet Khatri, Marco Cerezo, and Patrick J. Coles
New Journal of Physics, arXiv:1908.04416, LA-UR-19-28095
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2. Main Results
Noise resilience of Variational Quantum Compiling (VQC)
• Measurement noise (readout error).
• Incoherent gate noise and decoherence processes: Pauli
channels and non-unital Pauli channels.
Implementations of VQC on IBM’s noisy quantum simulator
• Quantum Fourier transform (QFT)
• Toffoli
• W-state preparation
Noise resilience in every case.
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3. Background
Applications of quantum computing:
• Quantum simulation [Fey99].
• Factoring [Sho99].
• Optimization [Gro96].
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4. Background
Noisy intermediate-scale quantum (NISQ) computers [Pre12]
Limitations of NISQ devices:
• Limited number of qubits.
• Limited connectivity between qubits.
• Restricted (hardware-specific) gate alphabets.
• Limited circuit depth due to noise.
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5. Background
Noisy intermediate-scale quantum (NISQ) computers [Pre12]
Limitations of NISQ devices:
• Limited number of qubits.
• Limited connectivity between qubits.
• Restricted (hardware-specific) gate alphabets.
• Limited circuit depth due to noise.
What can be done with NISQ devices?
• Error mitigation [TBG17, LJF+17, SCC18, MBJA+19].
• Variational hybrid quantum-classical algorithm [MRBAG16].
• Inherent noise resilience to certain noise models
[MRBAG16, SKCC20].
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6. Variational Quantum Compiling (VQC)
Quantum compiler
Conversion of an high-level algorithm into a lower-level machine
code [CFM17, HSST18, BDB+18].
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7. Variational Quantum Compiling (VQC)
Main goal of VQC [KLP+19, JB18, HSNF18]
Target unitary = U
Trainable unitary = V
• Full unitary matrix compiling:
Compile U by using V (k, α) that has approximately the same
action as U on any given input state.
α: continuous, k: discrete
• Fixed input state compiling:
U|ψ0 = V |ψ0 or U |0 = V |0 .
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8. Variational quantum compiling (VQC)
Applications of VQC:
• Algorithmic depth compression.
• Device specific compilation.
• Variational fast forwarding for quantum simulation [CHI+19].
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9. Quantum-Classical Hybrid Algorithm
Classical computers
• Optimization.
• Quantum simulation.
Noisy Quantum computers
• Quantum simulation.
• Optimization.
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10. Variational quantum compiling
Full unitary matrix compiling
Target unitary U
Trainable unitary V
Questions:
1. Cost function?
2. Task for a quantum computer?
3. Task for a classical computer?
4. How to perform optimization?
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11. Variational quantum compiling
Quantum Computer
Input: U
Output: Vkopt(αopt)
Continuous
parameter
optimizer over α
Structure
parameter
optimizer over k
Classical Computer
If α optimal
If α not optimal
Gate
sequence
Vk(α)
Cost
C(U, Vk(α))
|0 H •
U
• H
|0 H • • H
...
|0 H • • H
|0
V ∗
|0
...
|0
HST
|0 H •
U
• H
|0 H •
...
|0 H •
|0
V ∗
|0
...
|0
LHST
OR
OR
(a) Variable structure approach
⊕
⊕
⊕
⊕
Fixed
structure
Variable
structure
with L = 4
⊕
⊕
Optimal L = 5
compilation
(b) Fixed structure approach
Ansatz with
each layer
parameterized by
two-qubit gates
Two
layers
Figure 1: Outline for Full Unitary Matrix Compiling [KLP+19].
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12. Cost function for VQC
Lorem ipsum Lorem ipsum
Figure 2: a) The Hilbert-Schmidt Test (HST). b) The Local Hilbert-Schmidt Test
(LHST).
CHST = 1 − FHST = 1 − | Tr(V †
U)|2
/d2
,
CLHST = 1 − FLHST = 1 −
1
n
n
j=1
F
(j)
LHST.
CHST might exhibit barren plateau [MBS+18, CSV+20].
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13. Cost function for FUMC
Global Cost Function
CHST = 1 − FHST = 1 − | Tr(V †
U)|2
/d2
,
CHST =
d + 1
d
(1 − F(U, V )).
Faithfulness of CLHST:
CLHST ≤ CHST ≤ nCLHST.
• Cost functions are zero iff U = V (up to global phase).
• Efficient to compute on a quantum computer.
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14. Noise models
Pauli noise
• E(XkZl ) = ckl
XkZl .
• T2 process as a special case (dephasing channel).
• Depolarizing noise ρ → pρ + (1 − p)I/2n.
Non-unital Pauli noise
• E(I) = I + (k,l)=(0,0)
dkl
XkZl .
• For (k, l) = (0, 0), E(XkZl ) = ckl
XkZl .
• T1 process as a special case (amplitude damping channel).
Measurement noise
Ideal: {P0 = |0 0|, P1 = |1 1|}.
Noisy:{P0, P1} with P0 = p00|0 0| + p01|1 1| and
P1 = p10|0 0| + p11|1 1|, where p00 + p10 = 1, p01 + p11 = 1. 12 / 17

15. Main results: Optimal parameter resilience
Lorem ipsum
Figure 3: Noise Model 1. PGN = Pauli gate noise, MN = Measurement noise,
PN=Pauli noise, DN = depolarizing noise, NUPN = non-unital Pauli noise.
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16. Main results: Optimal parameter resilience
• Quantum circuit = QC.
• Cost function = CQC(V ).
• Noisy cost function (QC is run in the presence of some noise
process N) = CQC(V ).
Optimal solutions:
Vopt
d = {V ∈ Vd : CQC(V ) = min
V ∈Vd
CQC(V )} ,
Vopt
d = {V ∈ Vd : CQC(V ) = min
V ∈Vd
CQC(V )} .
CQC(V ) exhibits strong-optimal parameter resilience to N if
Vopt
d = Vopt
d .
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17. Main results: Optimal parameter resilience
Lorem ipsum
Figure 3: Noise Model 1. PGN = Pauli gate noise, MN = Measurement noise,
PN=Pauli noise, DN = depolarizing noise, NUPN = non-unital Pauli noise.
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18. Implementations
Figure 4: (a) The dressed CNOT is composed of a CNOT preceded and followed by
single-qubit gates Vk (αk ). (b) Two layers of the alternating-pair ansatz in the case of
four qubits. Each layer is composed of dressed CNOTs acting on alternating pairs of
neighboring qubits. (c) Schematic representation of the target-inspired ansatz. In this
approach, the gate sequence of dressed CNOTs is obtained from the gate sequence of
the target unitary U.
Vk(αk) = e−iαk,3σz /2
e−iαk,2σy /2
e−iαk,1σz /2
.
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19. Implementations
Figure 5: Quantum circuits for: (a) Toffoli Gate, (b) Three-qubit Quantum Fourier
Transform, and (c) Three-qubit W-state preparation. Here, Rm stands for the
controlled phase gate with a phase shift of φ = e2πi/2m
. For the three-qubit W-state
preparation circuit we have β1 = (2 arccos( 1/3), 0, 0) and β2 = (π/2, 0, 0).
Vk(αk) = e−iαk,3σz /2
e−iαk,2σy /2
e−iαk,1σz /2
.
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20. Implementations
Figure 6: VQC implementations: the Toffoli gate (top) and three-qubit QFT
(bottom). The ansatz for V (α) is: (a) one layer of the alternating-pair (AP) ansatz,
(b) two layers of the AP ansatz, (c) the target-inspired ansatz. The blue and green
curves respectively plot the values of CHST and CLHST obtained by noisy training. The
green and pink curves respectively plot the values of CHST and CLHST evaluated at the
variational parameters α obtained from the noisy optimization of V (α). The blue and
red dashed lines in (a) and (b) correspond to the minimum value of CHST and CLHST,
respectively, determined by optimizing V (α) in a noise-free environment.
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21. Conclusion
Noise resilience of Variational Quantum Compiling (VQC)
• Measurement noise (readout error).
• Incoherent gate noise and decoherence processes: Pauli
channels and non-unital Pauli channels.
Future directions
• Practically useful applications of VQC with a NISQ device.
• Noise resilience of other variational quantum algorithms.
Example: VQE.
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