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Quantum Computing Fundamentals via OO

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Carl Belle
Carl Belle

The document provides an introduction to quantum computing fundamentals using an object-oriented approach. It discusses quantum theory, registers, gates and simulations. Key concepts covered include superposition, matrix operations, single and multi-qubit gates like Pauli-X, CNOT and their representations. The presenter aims to demonstrate quantum computing principles via a .NET simulator called Q#.

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Quantum Computing Fundamentals via OO Carl Belle – carlyman@outlook.com ALT.NET Melbourne 2016
Thank You
About me VITAL STATISTICS Human Male Height: 183 cm Weight: 82 kg Favourite Enzyme: ATP Synthase PROFESSIONAL Bachelor of IT (RMIT) Research Student (RMIT): Computational Materials Discovery Senior Developer @ Xero
Me != Physicist Me != Mathematician
Agenda β€’ Introduction β€’ Theory – some maths will happen! β€’ Q# – a OO quantum computing simulator for .NET β€’ Futures
Why?
Why? The question: Can we simulate physics on a computer? The answer: Yes. But also no.
Hardware
Hardware Particle Isolated in a wafer Superconductor
Classical vs. Quantum Systems Absolutes vs. Probabilities
Classical vs. Quantum Systems CLASSICAL Manipulation of a register using Boolean algebra One operation = one result QUANTUM Manipulation of a register using Matrix algebra One operation = many results
Matrix Fundamentals Maths will shortly happen!
Matrix Multiplication π‘š Γ— 𝑛 Γ— 𝑛 Γ— 𝑝 = π‘š Γ— 𝑝 0 1 1 0 Γ— 0 1 0 Γ— 0 + (1 Γ— 1) 1 Γ— 0 + (0 Γ— 1) = 1 0
Matrix Tensor Multiplication 𝐴 ⨂ 𝐡 1 0 0 1 ⨂ 0 1 1 0 1 Γ— 0 1 1 0 0 Γ— 0 1 1 0 0 Γ— 0 1 1 0 1 Γ— 0 1 1 0 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0
The Register
The Register β€’ Constructed from qubits β€’ Consists of computational basis states β€’ Can be referred to as the β€˜state vector’ β€’ Preparation β€’ Normalisation
The Register - Superposition A quantum state added to another quantum state yields a further quantum state. Classical: 𝑦 = π‘₯ 16 = 16 β†’ 16 π‘π‘™π‘Žπ‘ π‘ π‘–π‘π‘Žπ‘™ 𝑏𝑖𝑑𝑠 Quantum: 𝑦 = 2 π‘₯ 16 = 24 β†’ 4 π‘žπ‘’π‘π‘–π‘‘π‘ 
1 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ πœ“ = 𝐴 0 + 𝐡 1
2 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ 𝑄1 = 𝐢 0 + 𝐷 1 ) β€’ πœ“ = 𝐴𝐢 00 + 𝐴𝐷 01 + 𝐡𝐢 10 + 𝐡𝐷|11⟩
3 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ 𝑄1 = 𝐢 0 + 𝐷 1 ) β€’ 𝑄2 = 𝐸 0 + 𝐹|1⟩) β€’ πœ“ = 𝐴𝐢𝐸 000 + 𝐴𝐢𝐹 001 + 𝐴𝐷𝐸 010 + 𝐴𝐷𝐹 011 + 𝐡𝐢𝐸 100 + 𝐡𝐢𝐹 101 + 𝐡𝐷𝐸 110 + 𝐡𝐷𝐹|111⟩
The Register – Normalisation The state vector which a quantum register represents has the following property: 𝛼 2 + 𝛽 2 + β‹― + 𝑛 2 = 1
Qubits More on Absolutes vs. Probabilities
Classical Bits vs. Quantum Bits CLASSICAL Absolutes QUANTUM Probabilities 𝐿𝑒𝑑 𝐴|0⟩ = 0.8, 𝐡|1⟩ = βˆ’0.6 0.8 2 + βˆ’0.6 2 = 0.64 + 0.36 = 1 β†’ 0 @ 64%, 1 @ 36% |1⟩ |0⟩
Gates Doing stuff with quantum things
Pauli-X Gate β€’ Acts on a single qubit β€’ Maps |0⟩ to |1⟩ and |1⟩ to |0⟩ β€’ Represented by the following matrix: 0 1 1 0
Pauli-X Gate πœ“ = 𝐴 0 + 𝐡 1 0 1 1 0 Γ— 𝐴|0⟩ 𝐡|1⟩ (0 Γ— 𝐴|0⟩) + (1 Γ— 𝐡|1⟩) (1 Γ— 𝐴|0⟩) + (0 Γ— 𝐡|1⟩) = 𝐡|1⟩ 𝐴|0⟩
Pauli-Z Gate β€’ Acts on a single qubit β€’ Leaves the basis state |0⟩ unchanged, and maps |1⟩ to βˆ’|1⟩ β€’ Represented by the following matrix: 1 0 0 βˆ’1
Pauli-Z Gate πœ“ = 𝐴 0 + 𝐡 1 1 0 0 βˆ’1 Γ— 𝐴|0⟩ 𝐡|1⟩ (1 Γ— 𝐴|0⟩) + ( 0 Γ— 𝐡|1⟩) (0 Γ— 𝐴|0⟩) + (βˆ’1 Γ— 𝐡|1⟩) = 𝐴|0⟩ βˆ’π΅|1⟩
Pauli-Y Gate β€’ Acts on a single qubit β€’ Maps |0⟩ to 𝑖|1⟩ and |1⟩ to βˆ’π‘–|0⟩ β€’ Represented by the following matrix: 0 βˆ’π‘– 𝑖 0
Imaginary Numbers β€’ The imaginary number 𝑖 has the following property: 𝑖2 = βˆ’1 β€’ When 𝑖 is squared, the sign of the term is changed β€’ Example: 5𝑖2 = 5 Γ— βˆ’1 = βˆ’5
Pauli-Y Gate πœ“ = 𝐴 0 + 𝐡 1 0 βˆ’π‘– 𝑖 0 Γ— 𝐴|0⟩ 𝐡|1⟩ (0 Γ— 𝐴|0⟩) + (βˆ’π‘– Γ— 𝐡|1⟩) (𝑖 Γ— 𝐴|0⟩) + ( 0 Γ— 𝐡|1⟩) = βˆ’π‘–π΅|1⟩ 𝑖𝐴|0⟩
SINGLE Qubit Gate, Multi qubit system β€’ State vector in a 2 qubit system contains 4 rows β€’ Single-qubit matrices contain 2 columns: not big enough to participate β€’ Single-qubit matrix must be β€˜stepped-up’ to be the appropriate size β€’ β€˜Step-up’ is achieved with the help of the identity matrix
Identity Matrix β€’ Simple diagonal matrix 1 0 0 1 β€’ Specifying the index of the qubit the gate is operating on β€’ Tensor operations with an identity matrix
Pauli-X Gate, 2 Qubit System 0 1 1 0 ⨂ 1 0 0 1 = 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ 0 Γ— 𝐴𝐢|00⟩ + 0 Γ— 𝐴𝐷|01⟩ + (1 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) (0 Γ— 𝐴𝐢|00⟩) + (0 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (1 Γ— 𝐡𝐷|11⟩) (1 Γ— 𝐴𝐢|00⟩) + (0 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) (0 Γ— 𝐴𝐢|00⟩) + (1 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) = 𝐡𝐢|00⟩ 𝐡𝐷|01⟩ 𝐴𝐢|10⟩ 𝐴𝐷|11⟩
Pauli-X Gate, 3 Qubit System (1) 1 0 0 1 ⨂ 0 1 1 0 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 ⨂ 1 0 0 1 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0
Pauli-X Gate, 3 Qubit System (2) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (1 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (1 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (1 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (1 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (1 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (1 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (1 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (1 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) = 𝐴𝐷𝐸|010⟩ 𝐴𝐷𝐹|011⟩ 𝐴𝐢𝐸|000⟩ 𝐴𝐢𝐹|001⟩ 𝐡𝐷𝐸|110⟩ 𝐡𝐷𝐹|111⟩ 𝐡𝐢𝐸|100⟩ 𝐡𝐢𝐹|101⟩ This protracted manual calculation is horrible, surely there must be a better way?
Hadamard Gate β€’ Acts on a single qubit β€’ Maps the basis state |0⟩ to (|0⟩ + |1⟩)/ 2 and |1⟩ to (|0⟩ βˆ’ |1⟩)/ 2 β€’ Represented by the following matrix: 1 2 1 1 1 βˆ’1
CNOT Gate β€’ Acts on two qubits β€’ Flips the target qubit if and only if the control qubit is 1 β€’ Singly-controlled NOT gate β€’ Represented by the following matrix: 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0
CNOT GATE, 2 Qubit System CNOT(0, 1) 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ = 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐷|11⟩ 𝐡𝐢|10⟩ CNOT(1, 0) 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ = 𝐴𝐢|00⟩ 𝐡𝐷|01⟩ 𝐡𝐢|10⟩ 𝐴𝐷|11⟩ Why are there two different CNOT matrices? Could it have anything to do with finding a nicer way of performing these calculations by hand?
CNOT Gate, 3 Qubit System CT 012 000 001 010 011 100 101 110 111 000 001 010 011 110 111 100 101 CT 012 ACE|000⟩ ACF|001⟩ ADE|010⟩ ADF|011⟩ BCE|100⟩ BCF|101⟩ BDE|110⟩ BDF|111⟩ β‡’ ACE|000⟩ ACF|001⟩ ADE|010⟩ ADF|011⟩ BDE|110⟩ BDF|111⟩ BCE|100⟩ BCF|101⟩ = ADE|000⟩ ADF|001⟩ ACE|010⟩ ACF|011⟩ BDE|100⟩ BDF|101⟩ BCE|110⟩ BCF|111⟩ Control qubit is 1, so the target qubit is flipped
CNOT Gate, 4 Qubit System CT 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADEG|0100⟩ ADEH|0101⟩ ADFG|0110⟩ ADFH|0111⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDEG|1100⟩ BDEH|1101⟩ BDFG|1110⟩ BDFH|1111⟩ CT β‡’ 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADFG|0110⟩ ADFH|0111⟩ ADEG|0100⟩ ADEH|0101⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDFG|1110⟩ BDFH|1111⟩ BDEG|1100⟩ BDEH|1101⟩ CT = 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADFG|0100⟩ ADFH|0101⟩ ADEG|0110⟩ ADEH|0111⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDFG|1100⟩ BDFH|1101⟩ BDEG|1110⟩ BDEH|1111⟩ Gee, that sure is a lot easier than that other method. The person who developed this must be truly amazing! And his personal hygiene must be second-to-none!
Differing CNOT Matrices – CNOT(0,1) 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 CT 01 AC|00⟩ AD|01⟩ BC|10⟩ BD|11⟩ AC AD BC BD AC AD BD BC β‡’ AC|00⟩ AD|01⟩ BD|11⟩ BC|10⟩ = AC|00⟩ AD|01⟩ BD|10⟩ BC|11⟩
Differing CNOT Matrices – CNOT(1,0) 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 TC 01 AC|00⟩ AD|01⟩ BC|10⟩ BD|11⟩ AC AD BC BD AC BD BC AD β‡’ AC|00⟩ BD|11⟩ BC|10⟩ AD|01⟩ = AC|00⟩ BD|01⟩ BC|10⟩ AD|11⟩
Tofolli Gate β€’ Acts on three qubits β€’ Flips the target qubit if and only if both control qubits are 1 β€’ Doubly-controlled NOT gate β€’ Represented by the following matrix: 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0
Measurement Did the thing work?
Measurement β€’ Measure in X β€’ Measure in Z β€’ Result is classical
OO Object-oriented quantuming!
Q# An object-oriented quantum computing simulator for .NET
Q# 1.0.0.4 β€’ An object-oriented quantum simulator written in C# β€’ Nuget package QSharp Mathematics β€’ Command parser β€’ Supports the following gates: CNOT, CSWAP, CZ, Hadamard, X, Y, Z, Swap, Toffoli β€’ Good for about 8 qubits
Q# Code Example 1 β€’ Create a register using 3 random qubits β€’ Print the initial state to the console
Q# Code Example 2 β€’ Create a register using 3 random qubits β€’ Print the initial state to the console β€’ Create a Pauli X gate β€’ Apply the X gate to qubit 0 β€’ Print the state to the console β€’ Apply the X gate to qubit 1 β€’ Print the final state to the console
Q# Code Example 3 β€’ Create a register using 5 random qubits β€’ Print the initial state to the console β€’ Read commands from a file β€’ Apply the commands using the Parser type β€’ Print the final state to the console β€’ Use the command results to generate a circuit diagram
Q# 1.0.0.5 – More Explosions! β€’ Rewritten matrix storage – huge memory savings! β€’ Supports custom Command types β€’ Good for 25 qubits (on my machine, anyway) β€’ Bigger, faster, more explosions! * β€’ Coming soon: β€’ .NET 1.0.0.5 β€’ C++ libraries β€’ Java libraries (maybe) β€’ * May (or may not) cause actual explosions
Remaining Challenges How long before we get our own quantum computers? And can we play Overwatch on them?
Remaining Challenges β€’ Hardware production β€’ Scaling β€’ Error correction β€’ Writing quantum software
Practical Quantum Computing What can I use it for?
D-Wave β€’ Controversial claims about β€˜quantumness’ β€’ Is it actually a quantum computer? β€’ Adiabatic device What an impressive beastie!
Practical Quantum Computing β€’ Quantum chemistry β€’ Complex molecule design and modelling β€’ Complex materials research β€’ Encryption and communication β€’ Searching large databases β€’ Solving complex systems of equations
Futures
Quantum Computing Futures β€’ Industry fragmentation β€’ Hardware production hurdles β€’ Large corporations (Microsoft, Google etc.) have their own quantum programs β€’ There is a lot of work to do!
Thank you very much

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Quantum Computing Fundamentals via OO

  • 1. Quantum Computing Fundamentals via OO Carl Belle – carlyman@outlook.com ALT.NET Melbourne 2016
  • 3. About me VITAL STATISTICS Human Male Height: 183 cm Weight: 82 kg Favourite Enzyme: ATP Synthase PROFESSIONAL Bachelor of IT (RMIT) Research Student (RMIT): Computational Materials Discovery Senior Developer @ Xero
  • 4. Me != Physicist Me != Mathematician
  • 5. Agenda β€’ Introduction β€’ Theory – some maths will happen! β€’ Q# – a OO quantum computing simulator for .NET β€’ Futures
  • 7. Why? The question: Can we simulate physics on a computer? The answer: Yes. But also no.
  • 9. Hardware Particle Isolated in a wafer Superconductor
  • 10. Classical vs. Quantum Systems Absolutes vs. Probabilities
  • 11. Classical vs. Quantum Systems CLASSICAL Manipulation of a register using Boolean algebra One operation = one result QUANTUM Manipulation of a register using Matrix algebra One operation = many results
  • 13. Matrix Multiplication π‘š Γ— 𝑛 Γ— 𝑛 Γ— 𝑝 = π‘š Γ— 𝑝 0 1 1 0 Γ— 0 1 0 Γ— 0 + (1 Γ— 1) 1 Γ— 0 + (0 Γ— 1) = 1 0
  • 14. Matrix Tensor Multiplication 𝐴 ⨂ 𝐡 1 0 0 1 ⨂ 0 1 1 0 1 Γ— 0 1 1 0 0 Γ— 0 1 1 0 0 Γ— 0 1 1 0 1 Γ— 0 1 1 0 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0
  • 16. The Register β€’ Constructed from qubits β€’ Consists of computational basis states β€’ Can be referred to as the β€˜state vector’ β€’ Preparation β€’ Normalisation
  • 17. The Register - Superposition A quantum state added to another quantum state yields a further quantum state. Classical: 𝑦 = π‘₯ 16 = 16 β†’ 16 π‘π‘™π‘Žπ‘ π‘ π‘–π‘π‘Žπ‘™ 𝑏𝑖𝑑𝑠 Quantum: 𝑦 = 2 π‘₯ 16 = 24 β†’ 4 π‘žπ‘’π‘π‘–π‘‘π‘ 
  • 18. 1 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ πœ“ = 𝐴 0 + 𝐡 1
  • 19. 2 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ 𝑄1 = 𝐢 0 + 𝐷 1 ) β€’ πœ“ = 𝐴𝐢 00 + 𝐴𝐷 01 + 𝐡𝐢 10 + 𝐡𝐷|11⟩
  • 20. 3 Qubit Register β€’ 𝑄0 = 𝐴 0 + 𝐡|1⟩) β€’ 𝑄1 = 𝐢 0 + 𝐷 1 ) β€’ 𝑄2 = 𝐸 0 + 𝐹|1⟩) β€’ πœ“ = 𝐴𝐢𝐸 000 + 𝐴𝐢𝐹 001 + 𝐴𝐷𝐸 010 + 𝐴𝐷𝐹 011 + 𝐡𝐢𝐸 100 + 𝐡𝐢𝐹 101 + 𝐡𝐷𝐸 110 + 𝐡𝐷𝐹|111⟩
  • 21. The Register – Normalisation The state vector which a quantum register represents has the following property: 𝛼 2 + 𝛽 2 + β‹― + 𝑛 2 = 1
  • 22. Qubits More on Absolutes vs. Probabilities
  • 23. Classical Bits vs. Quantum Bits CLASSICAL Absolutes QUANTUM Probabilities 𝐿𝑒𝑑 𝐴|0⟩ = 0.8, 𝐡|1⟩ = βˆ’0.6 0.8 2 + βˆ’0.6 2 = 0.64 + 0.36 = 1 β†’ 0 @ 64%, 1 @ 36% |1⟩ |0⟩
  • 24. Gates Doing stuff with quantum things
  • 25. Pauli-X Gate β€’ Acts on a single qubit β€’ Maps |0⟩ to |1⟩ and |1⟩ to |0⟩ β€’ Represented by the following matrix: 0 1 1 0
  • 26. Pauli-X Gate πœ“ = 𝐴 0 + 𝐡 1 0 1 1 0 Γ— 𝐴|0⟩ 𝐡|1⟩ (0 Γ— 𝐴|0⟩) + (1 Γ— 𝐡|1⟩) (1 Γ— 𝐴|0⟩) + (0 Γ— 𝐡|1⟩) = 𝐡|1⟩ 𝐴|0⟩
  • 27. Pauli-Z Gate β€’ Acts on a single qubit β€’ Leaves the basis state |0⟩ unchanged, and maps |1⟩ to βˆ’|1⟩ β€’ Represented by the following matrix: 1 0 0 βˆ’1
  • 28. Pauli-Z Gate πœ“ = 𝐴 0 + 𝐡 1 1 0 0 βˆ’1 Γ— 𝐴|0⟩ 𝐡|1⟩ (1 Γ— 𝐴|0⟩) + ( 0 Γ— 𝐡|1⟩) (0 Γ— 𝐴|0⟩) + (βˆ’1 Γ— 𝐡|1⟩) = 𝐴|0⟩ βˆ’π΅|1⟩
  • 29. Pauli-Y Gate β€’ Acts on a single qubit β€’ Maps |0⟩ to 𝑖|1⟩ and |1⟩ to βˆ’π‘–|0⟩ β€’ Represented by the following matrix: 0 βˆ’π‘– 𝑖 0
  • 30. Imaginary Numbers β€’ The imaginary number 𝑖 has the following property: 𝑖2 = βˆ’1 β€’ When 𝑖 is squared, the sign of the term is changed β€’ Example: 5𝑖2 = 5 Γ— βˆ’1 = βˆ’5
  • 31. Pauli-Y Gate πœ“ = 𝐴 0 + 𝐡 1 0 βˆ’π‘– 𝑖 0 Γ— 𝐴|0⟩ 𝐡|1⟩ (0 Γ— 𝐴|0⟩) + (βˆ’π‘– Γ— 𝐡|1⟩) (𝑖 Γ— 𝐴|0⟩) + ( 0 Γ— 𝐡|1⟩) = βˆ’π‘–π΅|1⟩ 𝑖𝐴|0⟩
  • 32. SINGLE Qubit Gate, Multi qubit system β€’ State vector in a 2 qubit system contains 4 rows β€’ Single-qubit matrices contain 2 columns: not big enough to participate β€’ Single-qubit matrix must be β€˜stepped-up’ to be the appropriate size β€’ β€˜Step-up’ is achieved with the help of the identity matrix
  • 33. Identity Matrix β€’ Simple diagonal matrix 1 0 0 1 β€’ Specifying the index of the qubit the gate is operating on β€’ Tensor operations with an identity matrix
  • 34. Pauli-X Gate, 2 Qubit System 0 1 1 0 ⨂ 1 0 0 1 = 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ 0 Γ— 𝐴𝐢|00⟩ + 0 Γ— 𝐴𝐷|01⟩ + (1 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) (0 Γ— 𝐴𝐢|00⟩) + (0 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (1 Γ— 𝐡𝐷|11⟩) (1 Γ— 𝐴𝐢|00⟩) + (0 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) (0 Γ— 𝐴𝐢|00⟩) + (1 Γ— 𝐴𝐷|01⟩) + (0 Γ— 𝐡𝐢|10⟩) + (0 Γ— 𝐡𝐷|11⟩) = 𝐡𝐢|00⟩ 𝐡𝐷|01⟩ 𝐴𝐢|10⟩ 𝐴𝐷|11⟩
  • 35. Pauli-X Gate, 3 Qubit System (1) 1 0 0 1 ⨂ 0 1 1 0 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 ⨂ 1 0 0 1 = 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0
  • 36. Pauli-X Gate, 3 Qubit System (2) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (1 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (1 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (1 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (1 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (1 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (1 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (1 Γ— 𝐡𝐢𝐸|100⟩) + (0 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) (0 Γ— 𝐴𝐢𝐸|000⟩) + (0 Γ— 𝐴𝐢𝐹|001⟩) + (0 Γ— 𝐴𝐷𝐸|010⟩) + (0 Γ— 𝐴𝐷𝐹|011⟩) + (0 Γ— 𝐡𝐢𝐸|100⟩) + (1 Γ— 𝐡𝐢𝐹|101⟩) + (0 Γ— 𝐡𝐷𝐸|110⟩) + (0 Γ— 𝐡𝐷𝐹|111⟩) = 𝐴𝐷𝐸|010⟩ 𝐴𝐷𝐹|011⟩ 𝐴𝐢𝐸|000⟩ 𝐴𝐢𝐹|001⟩ 𝐡𝐷𝐸|110⟩ 𝐡𝐷𝐹|111⟩ 𝐡𝐢𝐸|100⟩ 𝐡𝐢𝐹|101⟩ This protracted manual calculation is horrible, surely there must be a better way?
  • 37. Hadamard Gate β€’ Acts on a single qubit β€’ Maps the basis state |0⟩ to (|0⟩ + |1⟩)/ 2 and |1⟩ to (|0⟩ βˆ’ |1⟩)/ 2 β€’ Represented by the following matrix: 1 2 1 1 1 βˆ’1
  • 38. CNOT Gate β€’ Acts on two qubits β€’ Flips the target qubit if and only if the control qubit is 1 β€’ Singly-controlled NOT gate β€’ Represented by the following matrix: 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0
  • 39. CNOT GATE, 2 Qubit System CNOT(0, 1) 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ = 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐷|11⟩ 𝐡𝐢|10⟩ CNOT(1, 0) 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 Γ— 𝐴𝐢|00⟩ 𝐴𝐷|01⟩ 𝐡𝐢|10⟩ 𝐡𝐷|11⟩ = 𝐴𝐢|00⟩ 𝐡𝐷|01⟩ 𝐡𝐢|10⟩ 𝐴𝐷|11⟩ Why are there two different CNOT matrices? Could it have anything to do with finding a nicer way of performing these calculations by hand?
  • 40. CNOT Gate, 3 Qubit System CT 012 000 001 010 011 100 101 110 111 000 001 010 011 110 111 100 101 CT 012 ACE|000⟩ ACF|001⟩ ADE|010⟩ ADF|011⟩ BCE|100⟩ BCF|101⟩ BDE|110⟩ BDF|111⟩ β‡’ ACE|000⟩ ACF|001⟩ ADE|010⟩ ADF|011⟩ BDE|110⟩ BDF|111⟩ BCE|100⟩ BCF|101⟩ = ADE|000⟩ ADF|001⟩ ACE|010⟩ ACF|011⟩ BDE|100⟩ BDF|101⟩ BCE|110⟩ BCF|111⟩ Control qubit is 1, so the target qubit is flipped
  • 41. CNOT Gate, 4 Qubit System CT 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADEG|0100⟩ ADEH|0101⟩ ADFG|0110⟩ ADFH|0111⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDEG|1100⟩ BDEH|1101⟩ BDFG|1110⟩ BDFH|1111⟩ CT β‡’ 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADFG|0110⟩ ADFH|0111⟩ ADEG|0100⟩ ADEH|0101⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDFG|1110⟩ BDFH|1111⟩ BDEG|1100⟩ BDEH|1101⟩ CT = 0123 ACEG|0000⟩ ACEH|0001⟩ ACFG|0010⟩ ACFH|0011⟩ ADFG|0100⟩ ADFH|0101⟩ ADEG|0110⟩ ADEH|0111⟩ BCEG|1000⟩ BCEH|1001⟩ BCFG|1010⟩ BCFH|1011⟩ BDFG|1100⟩ BDFH|1101⟩ BDEG|1110⟩ BDEH|1111⟩ Gee, that sure is a lot easier than that other method. The person who developed this must be truly amazing! And his personal hygiene must be second-to-none!
  • 42. Differing CNOT Matrices – CNOT(0,1) 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 CT 01 AC|00⟩ AD|01⟩ BC|10⟩ BD|11⟩ AC AD BC BD AC AD BD BC β‡’ AC|00⟩ AD|01⟩ BD|11⟩ BC|10⟩ = AC|00⟩ AD|01⟩ BD|10⟩ BC|11⟩
  • 43. Differing CNOT Matrices – CNOT(1,0) 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 TC 01 AC|00⟩ AD|01⟩ BC|10⟩ BD|11⟩ AC AD BC BD AC BD BC AD β‡’ AC|00⟩ BD|11⟩ BC|10⟩ AD|01⟩ = AC|00⟩ BD|01⟩ BC|10⟩ AD|11⟩
  • 44. Tofolli Gate β€’ Acts on three qubits β€’ Flips the target qubit if and only if both control qubits are 1 β€’ Doubly-controlled NOT gate β€’ Represented by the following matrix: 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0
  • 46. Measurement β€’ Measure in X β€’ Measure in Z β€’ Result is classical
  • 48. Q# An object-oriented quantum computing simulator for .NET
  • 49. Q# 1.0.0.4 β€’ An object-oriented quantum simulator written in C# β€’ Nuget package QSharp Mathematics β€’ Command parser β€’ Supports the following gates: CNOT, CSWAP, CZ, Hadamard, X, Y, Z, Swap, Toffoli β€’ Good for about 8 qubits
  • 50. Q# Code Example 1 β€’ Create a register using 3 random qubits β€’ Print the initial state to the console
  • 51. Q# Code Example 2 β€’ Create a register using 3 random qubits β€’ Print the initial state to the console β€’ Create a Pauli X gate β€’ Apply the X gate to qubit 0 β€’ Print the state to the console β€’ Apply the X gate to qubit 1 β€’ Print the final state to the console
  • 52. Q# Code Example 3 β€’ Create a register using 5 random qubits β€’ Print the initial state to the console β€’ Read commands from a file β€’ Apply the commands using the Parser type β€’ Print the final state to the console β€’ Use the command results to generate a circuit diagram
  • 53. Q# 1.0.0.5 – More Explosions! β€’ Rewritten matrix storage – huge memory savings! β€’ Supports custom Command types β€’ Good for 25 qubits (on my machine, anyway) β€’ Bigger, faster, more explosions! * β€’ Coming soon: β€’ .NET 1.0.0.5 β€’ C++ libraries β€’ Java libraries (maybe) β€’ * May (or may not) cause actual explosions
  • 54. Remaining Challenges How long before we get our own quantum computers? And can we play Overwatch on them?
  • 55. Remaining Challenges β€’ Hardware production β€’ Scaling β€’ Error correction β€’ Writing quantum software
  • 56. Practical Quantum Computing What can I use it for?
  • 57. D-Wave β€’ Controversial claims about β€˜quantumness’ β€’ Is it actually a quantum computer? β€’ Adiabatic device What an impressive beastie!
  • 58. Practical Quantum Computing β€’ Quantum chemistry β€’ Complex molecule design and modelling β€’ Complex materials research β€’ Encryption and communication β€’ Searching large databases β€’ Solving complex systems of equations
  • 60. Quantum Computing Futures β€’ Industry fragmentation β€’ Hardware production hurdles β€’ Large corporations (Microsoft, Google etc.) have their own quantum programs β€’ There is a lot of work to do!