[Paper Review] Quantum Simulated Annealing

Somma, R., Rolando D. Somma+1 more|ArXiv.org|Dec 6, 2007

Quantum Computing Algorithms and Architecture4 references12 citations

TL;DR

This paper presents a quantum algorithm for combinatorial optimization that accelerates classical simulated annealing using quantum walks, phase estimation, and the quantum Zeno effect. It achieves a quadratic speedup over classical Markov chain Monte Carlo methods, reducing the number of required operations from $\mathcal{O}(1/\delta)$ to $\mathcal{O}(1/\sqrt{\delta})$, where $\delta$ is the spectral gap of the stochastic matrix, significantly improving performance for hard instances with small gaps.

ABSTRACT

We develop a quantum algorithm to solve combinatorial optimization problems through quantum simulation of a classical annealing process. Our algorithm combines techniques from quantum walks, quantum phase estimation, and quantum Zeno effect. It can be viewed as a quantum analogue of the discrete-time Markov chain Monte Carlo implementation of classical simulated annealing. Our implementation scales with the inverse of the square root of the minimum spectral gap of the stochastic matrix used in the classical simulation. The quantum algorithm outperforms the classical one, which scales with the inverse of the gap.

Motivation & Objective

• To develop a quantum algorithm that simulates classical simulated annealing more efficiently than classical Markov chain Monte Carlo (MCMC) methods.
• To address the inefficiency of classical simulated annealing when the spectral gap $\delta$ becomes exponentially small with problem size, which hampers convergence.
• To leverage quantum resources—specifically quantum walks, phase estimation, and the quantum Zeno effect—to achieve a provable speedup in solving combinatorial optimization problems.
• To demonstrate that quantum simulation of the annealing process can outperform classical counterparts, especially in cases where $\delta \ll 1$.

Proposed method

• The algorithm constructs a quantum walk from the transition matrix of a classical reversible Markov chain, using a similarity transformation to ensure unitarity and preserve spectral properties.
• It employs quantum phase estimation (QPE) to project the system onto the ground state of the annealing Hamiltonian at each inverse temperature $\beta_k$.
• The quantum Zeno effect is utilized by performing frequent projective measurements on ancilla qubits to suppress transitions to excited states, effectively slowing evolution and enhancing ground state projection.
• The algorithm proceeds through a sequence of inverse temperatures $\beta_1, \beta_2, \dots, \beta_Q = \beta_f$, applying QPE at each step to gradually cool the system toward the ground state.
• The final state is projected onto the all-zero ancilla register, and the probability of measuring the ground state is bounded using error terms from phase estimation and spectral gap considerations.
• The implementation complexity is derived by analyzing the cumulative error from multiple phase estimation steps and the decay of excited state amplitudes over time.

Experimental results

Research questions

• RQ1 Can quantum simulation of classical annealing processes yield a provable speedup over classical MCMC-based simulated annealing?
• RQ2 How does the use of quantum walks and phase estimation affect the convergence rate of the annealing process compared to classical methods?
• RQ3 To what extent does the quantum Zeno effect enhance the fidelity of ground state preparation in the quantum annealing framework?
• RQ4 What is the scaling of the number of operations required for quantum simulated annealing in terms of the spectral gap $\delta$?
• RQ5 Does the quantum algorithm maintain bounded error probability while reducing the number of state-generating operations compared to classical simulated annealing?

Key findings

• The quantum simulated annealing (QSA) algorithm achieves a complexity of $\mathcal{O}(\log^3(d/\epsilon^2)/(\sqrt{\delta}\epsilon^2))$ operations to find an optimal solution with error probability $\epsilon$, representing a quadratic speedup over classical simulated annealing.
• Classical simulated annealing requires $\mathcal{O}(1/\delta)$ operations, while the quantum version reduces this to $\mathcal{O}(1/\sqrt{\delta})$, making it significantly more efficient for problems with small spectral gaps.
• The error probability of the quantum algorithm is bounded above by $d e^{-\beta_f \gamma} + \tau' Q \nu^2$, where $\gamma$ is the spectral gap of the generator matrix and $\tau'$ is a constant, ensuring bounded failure probability.
• The use of repeated phase estimation with ancilla qubits and projective measurements enables effective implementation of the quantum Zeno effect, suppressing transitions to excited states and improving ground state fidelity.
• The algorithm's performance is most advantageous for hard instances where $\delta \ll 1$, as the quadratic dependence on $\sqrt{\delta}$ leads to dramatic reductions in required operations.
• The analysis shows that the final state is projected onto the ground state with high probability when the number of phase estimation steps $Q$ and the ancilla precision $p$ are chosen appropriately, with the error scaling controlled by $\nu^2$ and $Q$.