Classically Sampling Noisy Quantum Circuits in Quasi-Polynomial Time under Approximate Markovianity

TL;DR

A classical algorithm is presented that runs in $n^{\rm{polylog}(n)}$ time for simulating quantum circuits under local depolarizing noise, thereby ruling out their quantum advantage in these settings and suggesting that noise generically enforces approximate Markovianity and classical simulability.

Abstract

While quantum computing can accomplish tasks that are classically intractable, the presence of noise may destroy this advantage in the absence of fault tolerance. In this work, we present a classical algorithm that runs in $n^{\rm{polylog}(n)}$ time for simulating quantum circuits under local depolarizing noise, thereby ruling out their quantum advantage in these settings. Our algorithm leverages a property called approximate Markovianity to sequentially sample from the measurement outcome distribution of noisy circuits. We establish approximate Markovianity in a broad range of circuits: (1) we prove that it holds for any circuit when the noise rate exceeds a constant threshold, and (2) we provide strong analytical and numerical evidence that it holds for random quantum circuits subject to any constant noise rate. These regimes include previously known classically simulable cases as well as new ones, such as shallow random circuits without anticoncentration, where prior algorithms fail. Taken together, our results significantly extend the boundary of classical simulability and suggest that noise generically enforces approximate Markovianity and classical simulability, thereby highlighting the limitation of noisy quantum circuits in demonstrating quantum advantage.

Figures (7)

• Figure 1: Comparison with previous state-of-the-art classical simulation algorithms for noisy quantum circuits.
• Figure 2: (a) A one-dimensional noisy brickwork circuit with depth $d$. The double legs denote the bra and ket space. The backward light cone of a local observable $O$ is shaded in blue. (b,c) Schematics of our qudit-by-qudit sampling algorithm. (b) One possible sampling path on a two-dimensional grid. (c) The conditional distribution $P_{X_i|B(X_i, l) \cap X_{<i}}$ only depends on the qudits in the ball of radius $l$ around $X_i$ (region circled by the dashed line). We only include qudits in the ball that are already sampled.
• Figure 3: (a) After taking the Haar average, $k$ copies of the noisy random circuit map to an exponential sum of the permutation elements $\sigma$ and $\tau$ at each spacetime location, weighted by the Weingarten function (dashed line). The grey circles represent the depolarizing channel. (b) The statistical mechanics model after integrating out the $\tau$ variables. The boundaries are pinned to elements given in Proposition \ref{['prop:fourth_moment_partition']}.
• Figure 4: Numerical results for approximate Markovianity in noisy random quantum circuits. (a) $\bar{D}$ as a function of $l_{AC}$ at different noise rate $p$. (b) Inverse Markov length $1/\xi$ as a function of $p$. (c) $\bar{D}$ as a function of $l_{AC}$ at different depth $d$. (d) $1/\xi$ as a function of $d$.
• Figure 5: (a) A noisy brickwork circuit in one dimension. (b) The interaction graph associated with the circuit in (a).

Theorems & Definitions (53)

• Proposition 1: Rephrased from aharonov1996limitations
• Definition 1
• Proposition 2: Pinsker's inequality
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• Proposition 3
• Proposition 4
• Proposition 5: Informal