Quantum Simulation on theQuantum Hardware
Hidetoshi Nishimori
Tokyo Institute of Technology
iTHEMS RIKEN
Tohoku UniversityQuantum simulation
Richard Feynman (1981)“I want to talk about the possibility that there is to be an exact
simulation, that the computer will do exactly the same as nature.”••••(my) Definition of quantum simulation
To reproduce the behavior of a quantum-mechanical system on an
artificial quantum device.

Advantage over real experiment
Precise control microscopic parameters. E.g. interactions between spins

Disadvantage
Difficult to build large systems without noise.

Today’s topic
Successfully mitigated the difficulty in a (relatively) large system with up
to several thousand qubits.

Qubits (quantum bits) in quantum device
0,100, 01, 10, 11000, 001, 010, 011
100, 101, 110, 111

Tiny
superconducting
ring at low T

0000, 0001, 0010, 0011
0100, 0101, 0110, 0111
1000, 1001, 1010, 1011
1100, 1101, 1110, 1111

00000, 00001, 00010, 00011
00100, 00101, 00110, 00111
01000, 01001, 01010, 01011
01100, 01101, 01110, 01111
10000, 10001, 10010, 10011
10100, 10101, 10110, 10111
11000, 11001, 11010, 11011
11100, 11101, 11110, 11111

000000, 000001, 000010, 000011
000100, 000101, 000110, 000111
001000, 001001, 001010, 001011
001100, 001101, 001110, 001111
010000, 010001, 010010, 010011
010100, 010101, 010110, 010111
011000, 011001, 011010, 011011
011100, 011101, 011110, 011111
100000, 100001, 100010, 100011
100100, 100101, 100110, 100111
101000, 101001, 101010, 101011
101100, 101101, 101110, 101111
110000, 110001, 110010, 110011
110100, 110101, 110110, 110111
111000, 111001, 111010, 111011
111100, 111101, 111110, 111111

Superposition ofvery many statesTypes of hardware for quantum simulation•

Large-scale fault-tolerant quantum computer (FTQC)
The “ideal” quantum computer to be realized in the (far) future.
Theoretically known to be effective for solving some hard problems efficiently.
Challenge : To build a device of sufficient scale to realize the “ideal” goal.

•NISQ (Noisy Intermediate-Scale Quantum computer)
Already realized in small scales.
Small-scale experiments have been conducted. Still within the reach of classical computers.
Challenge: Significant reduction of noise is needed for meaningful results.

•

Quantum annealing and quantum simulator
Already realized in intermediate scales.
Simulations have already been realized with significant scale (to be shown later).
Challenge: Limited in the type of systems to be simulated, e.g. the transverse-field Ising model.

Quantum annealing
T. Kadowaki and H. Nishimori (1998)

Generic quantum algorithm for combinatorial-optimization problems- To find the ground (lowest energy) state of the Ising model with complex interactions (spin glass)- Many problems of practical importance are formulated as the Ising problem.
- Examples: optimal vehicle routing, job scheduling, portfolio optimization, fault detection

EnergyTo find the state with the lowest energy
in a very rugged energy landscape.

00000, 00001, 00010, 00011 00100, 00101, 00110, 00111

2N states

Quantum annealing
T. Kadowaki and H. Nishimori (1998)

Generic quantum algorithm for combinatorial-optimization problems

MethodAppropriate control of coefficients of the transverse-field Ising modelTime evolution

Superpositionof all states(trivial)

00000, 00001, 00010, 00011
00100, 00101, 00110, 00111
01000, 01001, 01010, 01011
01100, 01101, 01110, 01111
10000, 10001, 10010, 10011
10100, 10101, 10110, 10111
11000, 11001, 11010, 11011
11100, 11101, 11110, 11111

10110

Solution (non-trivial)

Energy- To find the ground state of Ising model with complex interactions (spin glass)- Many problems of practical importance are formulated this way.
- Example: optimal vehicle routing, job scheduling, portfolio optimization, fault detection

state

01

10

1

10000

1

1

Choose the optimal state out of 212=4096 states

Representation of the Ising model on the device
QPU•The Ising model representing the problem can be implemented on the chip.

•The size is limited to 5300 qubits on the current model.

Examples of quantum simulation by quantum annealing•
••
•
•••

Static properties

Spin glass
Kosterlitz-Thouless transition
Spin ice
Shastry-Sutherland model
Scalar field theory
Griffiths-McCoy singularity

Harris et al, Science (2018)
King et al, Nature (2018)
King et al, Science (2021)
Kairys et al, PRX Quantum (2020)
Abel et al, PRX Quantum (2021)
Nishimura et al, Phys. Rev. A (2020)

Dynamical properties

Kibble-Zurek mechanism

Gardas et al, Sci. Rep. (2018)
Weinberg et al, Phys. Rev. Lett. (2020)
Bando et al, Phys. Rev. Res. (2020)•Frustrated spin chain

King et al, Nature Phys. (2022)
King et al, PRX Quantum (2021)

Quantum simulation of the Kibble-Zurek mechanism

Phys. Rev. Res.
2, 033369 (2020)

Nature Phys. 18, 1324 (2022)

Phase transition (classical)
2d Ising modelLow temperature (ordered)

Critical temperature

High temperature (disordered)Temperature decrease
Slow decrease of temperature:

System traces equilibrium.

Fast decrease of temperature :Falls out of equilibrium. Defects are created at T=0.

Problem: Relation of the density of defects ρ and the time scale ta of temperature decrease ρ = f (ta)

1d quantum Ising model

Solution01
Critical point+111111

Defect (Kink)

Slow change

Γ/JSuperposition of all states↑1 ↑ 2 ↑3  ↑ L + ↑1 ↑ 2 ↑3  ↓ L +  + ↓1 ↓ 2 ↓3  ↓ L
00000, 00001, 00010, 00011
00100, 00101, 00110, 00111
01000, 01001, 01010, 01011
01100, 01101, 01110, 01111
10000, 10001, 10010, 10011
10100, 10101, 10110, 10111
11000, 11001, 11010, 11011
11100, 11101, 11110, 11111

By decreasing Γ/J at a finite rate, defects are created at Γ/J=0.

What is the number (density) of defects ρ as a function of annealing time ta? ρ = f (ta)Kibble-Zurek mechanism

1d (chain) problem
Number (density) of defects is known
exactly by quantum mechanics (noise-free).

Goal: To verify this formula on the D-Wave quantum annealer.

Evidence of thermal noise on the device● : Output from D-Wave

Bold lines: Quantum theory for finite temperature● : Classical theory

Agrees with the finite-temperature quantum theory.
S. Boixo et al, Nature Comm. 7, 10327 (2016)

Signal synchronization

Chip size is 1cm.

Light speed is 30cm/ns. Smallest experimental time scale: 2 nano second.
⇒
Signal synchronization should be carefully tuned to reach all qubits simultaneously.

Result• Data agrees with theory at short time scale up to 50 ns.
• The device runs coherently without noise.
• Kibble-Zurek theory has been confirmed including thecoefficient.

ρSignificanceta (nano sec)

• First case to run a large-scale artificial quantum
device with L=2000 qubits noise-free.
(cf. The largest quantum simulation on otherplatforms is about 200 qubits.)• Opened a path toward large-scale quantum
simulations beyond the capacity of classicalcomputers.

Quantum simulation of the 3D spin glass
King et al, arXiv:2207.13800

Theoryslope

Experiment

Realized 3d spin glass of
size 15×15×12

Coupling strength• Quantum simulation of 3d spin glass dynamics with 5300 qubits.
• Small deviations from the theory.
• Faster dynamics toward optimal by quantum than classical.

Conclusion• Large-scale quantum simulations have been conducted on a device with several thousandqubits.
• Confirmed that the device operates coherently (quantum-mechanically without noise) up
to about 20-50 nanosecond.
• For optimization:
Quality of data may be improved by running quantum annealing at short time scales
and/or with improved coherence.

Rapidly expanding field of quantum annealing
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Number of papers with keyword “quantum annealing”