We analyze the complexity of classically simulating continuous-time dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We prove that a polynomial time classical algorithm can be used to sample from the state of the spins when the rate of noise is higher than a threshold determined by the strength of the local interactions. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of spin systems arranged on two or higher dimensional grids, we show that for several noise channels, the problem of weakly simulating the output state of both purely Hamiltonian and purely dissipative dynamics is expected to be hard in the low-noise regime.We analyze the complexity of classically simulating continuous-time dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We prove that a polynomial time classical algorithm can be used to sample from the state of the spins when the rate of noise is higher than a threshold determined by the strength of the local interactions. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of spin systems arranged on two or higher dimensional grids, we show that for several noise channels, the problem of weakly simulating the output state of both purely Hamiltonian and purely dissipative dynamics is expected to be hard in the low-noise regime.

arXiv:2110.10638Oct 20, 20210 citations

Simulatability of locally-interacting open quantum spin systems

Rahul Trivedi, J. I. Cirac

We analyze the complexity of classically simulating the dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We show that for a system with $n$ spins, a polynomial time classical algorithm can be used to sample within a $O(1/\text{poly}(n))$ total-variation distance of the state of the spins when the rate of noise is sufficiently high. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of open quantum spin systems, we show that for certain physically relevant channels, the problem of sampling from the output state is BQP-complete in the low noise regime. Our results rigorously show the presence of phase transitions in the classical simulatability of the continuous time dynamics of open quantum spin systems.We analyze the complexity of classically simulating the dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We show that for a system with $n$ spins, a polynomial time classical algorithm can be used to sample within a $O(1/\text{poly}(n))$ total-variation distance of the state of the spins when the rate of noise is sufficiently high. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of open quantum spin systems, we show that for certain physically relevant channels, the problem of sampling from the output state is BQP-complete in the low noise regime. Our results rigorously show the presence of phase transitions in the classical simulatability of the continuous time dynamics of open quantum spin systems.

Oct 20, 20211 citation

Quantum-to-Classical Computability Transition via Negative Markov Chains

Hugo Lóio, Jacopo De Nardis, Tony Jin

We develop a recently introduced representation of quantum dynamics based on sampling negative Markov chain processes. By introducing particles and antiparticles, this formalism maps generic quantum dynamics onto a Markov process defined over an exponentially large configuration space. Within this framework, quantum complexity arises from the proliferation of stochastic particles, which ultimately renders classical simulation and sampling intractable beyond a certain timescale. In the presence of noise, we demonstrate that for any unitary evolution generated by a linear combination of local or pairwise interactions, there exists at least one noise channel that effectively classicalizes the system by suppressing particle growth and making Monte Carlo sampling efficient. As a corollary, we show that for this class of unitaries, the dynamics of an open quantum spin chain subject to depolarizing noise undergoes an exact transition to classical simulability once the noise strength exceeds a critical threshold which can be efficiently determined for any model.We develop a recently introduced representation of quantum dynamics based on sampling negative Markov chain processes. By introducing particles and antiparticles, this formalism maps generic quantum dynamics onto a Markov process defined over an exponentially large configuration space. Within this framework, quantum complexity arises from the proliferation of stochastic particles, which ultimately renders classical simulation and sampling intractable beyond a certain timescale. In the presence of noise, we demonstrate that for any unitary evolution generated by a linear combination of local or pairwise interactions, there exists at least one noise channel that effectively classicalizes the system by suppressing particle growth and making Monte Carlo sampling efficient. As a corollary, we show that for this class of unitaries, the dynamics of an open quantum spin chain subject to depolarizing noise undergoes an exact transition to classical simulability once the noise strength exceeds a critical threshold which can be efficiently determined for any model.

quant-pharXiv:2604.19889Apr 23, 20261 citation

The hardness of quantum spin dynamics

Chae-Yeun Park, Pablo A. M. Casares, Juan Miguel Arrazola +1 more

Recent experiments demonstrated quantum computational advantage in random circuit sampling and Gaussian boson sampling. However, it is unclear whether these experiments can lead to practical applications even after considerable research effort. On the other hand, simulating the quantum coherent dynamics of interacting spins has been considered as a potential first useful application of quantum computers, providing a possible quantum advantage. Despite evidence that simulating the dynamics of hundreds of interacting spins is challenging for classical computers, concrete proof is yet to emerge. We address this problem by proving that sampling from the output distribution generated by a wide class of quantum spin Hamiltonians is a hard problem for classical computers. Our proof is based on the Taylor series of the output probability, which contains the permanent of a matrix as a coefficient when bipartite spin interactions are considered. We devise a classical algorithm that extracts the coefficient using an oracle estimating the output probability. Since calculating the permanent is #P-hard, such an oracle does not exist unless the polynomial hierarchy collapses. With an anticoncentration conjecture, the hardness of the sampling task is also proven. Based on our proof, we estimate that an instance involving about 200 spins will be challenging for classical devices but feasible for intermediate-scale quantum computers with fault-tolerant qubits.Recent experiments demonstrated quantum computational advantage in random circuit sampling and Gaussian boson sampling. However, it is unclear whether these experiments can lead to practical applications even after considerable research effort. On the other hand, simulating the quantum coherent dynamics of interacting spins has been considered as a potential first useful application of quantum computers, providing a possible quantum advantage. Despite evidence that simulating the dynamics of hundreds of interacting spins is challenging for classical computers, concrete proof is yet to emerge. We address this problem by proving that sampling from the output distribution generated by a wide class of quantum spin Hamiltonians is a hard problem for classical computers. Our proof is based on the Taylor series of the output probability, which contains the permanent of a matrix as a coefficient when bipartite spin interactions are considered. We devise a classical algorithm that extracts the coefficient using an oracle estimating the output probability. Since calculating the permanent is #P-hard, such an oracle does not exist unless the polynomial hierarchy collapses. With an anticoncentration conjecture, the hardness of the sampling task is also proven. Based on our proof, we estimate that an instance involving about 200 spins will be challenging for classical devices but feasible for intermediate-scale quantum computers with fault-tolerant qubits.

arXiv:2312.07658Dec 12, 20231 citation

Optimized Trajectory Unraveling for Classical Simulation of Noisy Quantum Dynamics

Zhuo Chen, Yimu Bao, Soonwon Choi

The dynamics of open quantum systems can be simulated by unraveling it into an ensemble of pure state trajectories undergoing nonunitary monitored evolution, which has recently been shown to undergo measurement-induced entanglement phase transition. Here, we show that, for an arbitrary decoherence channel, one can optimize the unraveling scheme to lower the threshold for entanglement phase transition, thereby enabling efficient classical simulation of the open dynamics for a broader range of decoherence rates. Taking noisy random unitary circuits as a paradigmatic example, we analytically derive the optimum unraveling basis that on average minimizes the threshold. Moreover, we present a heuristic algorithm that adaptively optimizes the unraveling basis for given noise channels, also significantly extending the simulatable regime. When applied to noisy Hamiltonian dynamics, the heuristic approach indeed extends the regime of efficient classical simulation based on matrix product states beyond conventional quantum trajectory methods. Finally, we assess the possibility of using a quasi-local unraveling, which involves multiple qubits and time steps, to efficiently simulate open systems with an arbitrarily small but finite decoherence rate.The dynamics of open quantum systems can be simulated by unraveling it into an ensemble of pure state trajectories undergoing nonunitary monitored evolution, which has recently been shown to undergo measurement-induced entanglement phase transition. Here, we show that, for an arbitrary decoherence channel, one can optimize the unraveling scheme to lower the threshold for entanglement phase transition, thereby enabling efficient classical simulation of the open dynamics for a broader range of decoherence rates. Taking noisy random unitary circuits as a paradigmatic example, we analytically derive the optimum unraveling basis that on average minimizes the threshold. Moreover, we present a heuristic algorithm that adaptively optimizes the unraveling basis for given noise channels, also significantly extending the simulatable regime. When applied to noisy Hamiltonian dynamics, the heuristic approach indeed extends the regime of efficient classical simulation based on matrix product states beyond conventional quantum trajectory methods. Finally, we assess the possibility of using a quasi-local unraveling, which involves multiple qubits and time steps, to efficiently simulate open systems with an arbitrarily small but finite decoherence rate.

arXiv:2306.17161Physical Review LettersDec 6, 202412 citations

A Polynomial-Time Classical Algorithm for Noisy Quantum Circuits

Thomas Schuster, Chao Yin, Xun Gao +1 more

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.

arXiv:2407.12768Physical Review XAug 15, 202526 citations

We analyze the complexity of classically simulating the dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We show that for a system with

n

spins, a polynomial time classical algorithm can be used to sample within a

O(1/\text{poly}(n))

total-variation distance of the state of the spins when the rate of noise is sufficiently high. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of open quantum spin systems, we show that for certain physically relevant channels, the problem of sampling from the output state is BQP-complete in the low noise regime. Our results rigorously show the presence of phase transitions in the classical simulatability of the continuous time dynamics of open quantum spin systems.