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Experimental Realization of the Markov Chain Monte Carlo Algorithm on a Quantum Computer

Baptiste Claudon, Sergi Ramos-Calderer, Jean-Philip Piquemal

TL;DR

This work experimentally uses encodings of Markov chains to prepare quantum states and runs a quantum Markov Chain Monte Carlo algorithm (qMCMC) on Quantinuum's H2 and Helios quantum computers, demonstrating that it is possible to obtain accurate results on current Noisy Intermediate Scale Quantum (NISQ) hardware.

Abstract

Quantum algorithms present a quadratically improved complexity over classical ones for certain sampling tasks. For instance, the Quantum Amplitude Estimation (QAE) algorithm promises to speedup the estimation of the mean of certain functions, given access to the quantum state corresponding to the probability distribution to be sampled from. Classically, samples are often obtained by running steps a Markov Chain. In this work, we experimentally use encodings of Markov chains to prepare quantum states and run a quantum Markov Chain Monte Carlo algorithm (qMCMC) on Quantinuum's H2 and Helios quantum computers. We demonstrate that it is possible to obtain accurate results on current Noisy Intermediate Scale Quantum (NISQ) hardware, operating directly on the physical qubits.

Experimental Realization of the Markov Chain Monte Carlo Algorithm on a Quantum Computer

TL;DR

This work experimentally uses encodings of Markov chains to prepare quantum states and runs a quantum Markov Chain Monte Carlo algorithm (qMCMC) on Quantinuum's H2 and Helios quantum computers, demonstrating that it is possible to obtain accurate results on current Noisy Intermediate Scale Quantum (NISQ) hardware.

Abstract

Quantum algorithms present a quadratically improved complexity over classical ones for certain sampling tasks. For instance, the Quantum Amplitude Estimation (QAE) algorithm promises to speedup the estimation of the mean of certain functions, given access to the quantum state corresponding to the probability distribution to be sampled from. Classically, samples are often obtained by running steps a Markov Chain. In this work, we experimentally use encodings of Markov chains to prepare quantum states and run a quantum Markov Chain Monte Carlo algorithm (qMCMC) on Quantinuum's H2 and Helios quantum computers. We demonstrate that it is possible to obtain accurate results on current Noisy Intermediate Scale Quantum (NISQ) hardware, operating directly on the physical qubits.
Paper Structure (9 sections, 1 theorem, 14 equations, 9 figures, 11 tables)

This paper contains 9 sections, 1 theorem, 14 equations, 9 figures, 11 tables.

Table of Contents

  1. Quantum Markov Chain Monte Carlo
  2. Results
  3. State preparation and amplitude estimation
  4. Dual Space quantum Markov Chain Monte Carlo
  5. Discussion
  6. Acknowledgments
  7. References
  8. Circuits
  9. Data

Key Result

Theorem 1

Let $(U, \square)$ be a SPUE of an operator $A$, with qubitized walk operator $\mathcal{W}$. Let $\lambda\in ]-1, 1[$ be an eigenvalue of $A$ with eigenvector $\ket v$. Define $\theta=\cos^{-1}(\lambda)$. Then $\mathcal{W}$ has eigenvalues $e^{\pm i\theta}$ with eigenvectors supported on the two-dim

Figures (9)

  • Figure 1: Controlled-$\mathcal{W}$ circuit. $c$ labels the control qubit, $a$ the ancilla qubit and $x$ the state space qubit.
  • Figure 2: State preparation circuit. $c$ labels the control qubit, $a$ the ancilla qubit and $x$ the state space qubit. If the control qubit is measured in state $\ket0$, the ancilla qubit exits in state $\ket0$ and the state qubit in state $\ket\pi$.
  • Figure 3: Quantum circuit implementing the unitary $O$ in Szegedy's quantization procedure.
  • Figure 4: Measurement statistics of $10^4$ measurements of $V\ket0$ and $\mathcal{W} V\ket0$ on Quantinuum's H2-1, H2-2 and Helios devices. We expect $V\ket0$ to be a $1$ eigenvector of $\mathcal{W}$, thus should remain unchanged after the application of the walk operator. The eigenstate is $V\ket0=\mathcal{W}V\ket0=\frac{1}{\sqrt8}\left(\ket{10}+\ket{12}+\ket{18}+\ket{20}+\ket{42}+\ket{44}+\ket{50}+\ket{52}\right).$
  • Figure 5: State preparation circuit for the Linear Combination of Unitaries method.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Theorem 1