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Apr 15, 2023 297 likes •498 views

Classical simulations of quantum circuits Resource-theoretic approach to quantum computing Kamil Korzekwa Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Poland Outline 1. Motivation 2. Background 3.

SLIDE 1

Kamil Korzekwa

Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Poland

Classical simulations of quantum circuits

Resource-theoretic approach to quantum computing

SLIDE 2

K.K. (UJ)

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Kraków, 21/01/2020 Classical simulations of quantum circuits 2/18

Outline

S. Bartlett
H. Pashayan

In collaboration with:

SLIDE 3

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 3/18

Motivation

Foundations Applications Characterization, verification, and validation of near-term quantum devices Strong evidence that quantum computing is more powerful than classical computing. What component of quantum theory is responsible for this quantum speed-up?

Entanglement?
Coherence?
Contextuality?
Wigner negativity?
Special combination of the above?

SLIDE 4

K.K. (UJ)

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Kraków, 21/01/2020 Classical simulations of quantum circuits 4/18

Outline

a. (Qu)bits b. Universal sets of (quantum) gates c. Simulating quantum circuits

SLIDE 5

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 5/18

Background: (Qu)bits

E.g. 01 or 11

SLIDE 6

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 6/18

Background: Universal sets of (quantum) gates

2-qubit gate: CNOT General 1-qubit gate

SLIDE 7

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 7/18

Background: Simulating quantum circuits

Strong simulation Weak simulation Our simulation

SLIDE 8

K.K. (UJ)

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Kraków, 21/01/2020 Classical simulations of quantum circuits 8/18

Outline

a. Pauli gates and stabiliser states b. Clifford gates and Gottesmann-Knill c. Step 1: Gadgetizing T gates d. Step 2: Stabilizer decompositon e. Step 3: Sampling stabilizers f. Step 4: Fast norm estimation

SLIDE 9

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 9/18

Simulating Clifford + T circuits Pauli gates and stabiliser states

1-qubit Pauli gates: n-qubit Pauli gates: n-qubit stabilizer state: simultaneous eigenstate of n commuting Pauli matrices

SLIDE 10

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 10/18

Simulating Clifford + T circuits Clifford gates and Gottesmann-Knill theorem

Generators: CNOT Gottesmann-Knill theorem: evolution of stabiliser states through Clifford circuits can be efficiently described on a classical computer. (n-qubit stabiliser state described by n Pauli operators, each of them is mapped by a Clifford gate to another Pauli operator. Just keep track of stabilisers.)

SLIDE 11

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 11/18

Simulating Clifford + T circuits Clifford gates and Gottesmann-Knill theorem

Clifford gates are not universal! Adding a single T gate is enough!

With arbitrary accuracy (general circuit) (Clifford+T circuit)

SLIDE 12

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 12/18

Simulating Clifford + T circuits Step 1: Gadgetizing T gates with magic states

(Clifford+T circuit) (Clifford circuit) Precisely arXiv:1601.07601

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K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 13/18

Simulating Clifford + T circuits Step 2: Stabilizer decomposition of magic states

SLIDE 14

K.K. (UJ)

Use Gottesmann-Knill to evolve each term
Kraków, 21/01/2020

Classical simulations of quantum circuits 14/18

Simulating Clifford + T circuits Step 3: Sampling from stabilizer decomposition

(Clifford circuit) arXiv:1601.07601

SLIDE 15

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 15/18

Simulating Clifford + T circuits Step 4: Fast norm estimation

Employ the efficient stabilizer norm estimation from arXiv:1601.07601 Final run-time of the algorithm:

SLIDE 16

K.K. (UJ)

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Kraków, 21/01/2020 Classical simulations of quantum circuits 16/18

Outline

SLIDE 17

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 17/18

Unified simulation framework

Various splittings into free (efficiently simulable) theory and resourceful (exponentially hard to simulate) operations:

Clifford + T gates
Gaussian gates + Non-gaussian gate
Matchgate circuits + SWAP gate
…

Gadgetization Decomposition of resource states into free states Sampling from free- state decomposition Estimating probability

SLIDE 18

K.K. (UJ) Kraków, 21/01/2020 Classical simulations of quantum circuits 18/18

Outlook

New Quantum Resource Group established at Jagiellonian University (leader + 2 post-docs + 2 PhD students + MSc student) Objective 1: A unified framework for classical simulations of quantum circuits

1. Developing a unified scheme for classical simulation of universal quantum

circuits based on a three-step algorithm.

2. Devising novel algorithms with improved run-time scaling by employing

alternative free element decompositions (e.g. pure free states). Implementing these algorithms on classical computers and employing them to certify and verify NISQ devices.

3. Investigating the interconversion problem for the resource theory of magic

Kamil Korzekwa

Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Poland

Classical simulations of quantum circuits

Resource-theoretic approach to quantum computing

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Outline

Motivation

Foundations Applications Characterization, verification, and validation of near-term quantum devices Strong evidence that quantum computing is more powerful than classical computing. What component of quantum theory is responsible for this quantum speed-up?

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Outline

a. (Qu)bits b. Universal sets of (quantum) gates c. Simulating quantum circuits

Background: (Qu)bits

E.g. 01 or 11

Background: Universal sets of (quantum) gates

2-qubit gate: CNOT General 1-qubit gate

Background: Simulating quantum circuits

Strong simulation Weak simulation Our simulation

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Outline

a. Pauli gates and stabiliser states b. Clifford gates and Gottesmann-Knill c. Step 1: Gadgetizing T gates d. Step 2: Stabilizer decompositon e. Step 3: Sampling stabilizers f. Step 4: Fast norm estimation

Simulating Clifford + T circuits Pauli gates and stabiliser states

1-qubit Pauli gates: n-qubit Pauli gates: n-qubit stabilizer state: simultaneous eigenstate of n commuting Pauli matrices

Simulating Clifford + T circuits Clifford gates and Gottesmann-Knill theorem

Simulating Clifford + T circuits Clifford gates and Gottesmann-Knill theorem

Clifford gates are not universal! Adding a single T gate is enough!

Simulating Clifford + T circuits Step 1: Gadgetizing T gates with magic states

Simulating Clifford + T circuits Step 2: Stabilizer decomposition of magic states

Simulating Clifford + T circuits Step 3: Sampling from stabilizer decomposition

Simulating Clifford + T circuits Step 4: Fast norm estimation

Employ the efficient stabilizer norm estimation from arXiv:1601.07601 Final run-time of the algorithm:

1. Motivation 2. Background 3. Simulating Clifford + T circuits 4. Unified simulation framework 5. Outlook

Outline

Unified simulation framework

Various splittings into free (efficiently simulable) theory and resourceful (exponentially hard to simulate) operations:

Gadgetization Decomposition of resource states into free states Sampling from free- state decomposition Estimating probability

Outlook

New Quantum Resource Group established at Jagiellonian University (leader + 2 post-docs + 2 PhD students + MSc student) Objective 1: A unified framework for classical simulations of quantum circuits

circuits based on a three-step algorithm.

alternative free element decompositions (e.g. pure free states). Implementing these algorithms on classical computers and employing them to certify and verify NISQ devices.

states.