Efficient classical simulation of random shallow 2D quantum circuits

TL;DR

It is proved by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates.

Abstract

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied.

Figures (18)

• Figure 1: Schematic depiction of SEBD acting on a square lattice with circuit depth $d$. In all figures, the 2D circuit is depicted as a spacetime volume, with time flowing upwards. The green (respectively dotted blue) region denotes unmeasured (measured) qudits. In (a), we apply all gates in the lightcone of column 1, namely, those gates intersecting the spacetime volume shaded red. In (b), we simulate the computational basis measurement of column 1. In (c), we apply all gates in the lightcone of column 2 that were previously unperformed. Figure (d) depicts the algorithm at an intermediate stage of the simulation, after the measurements of about half of the qudits have been simulated. The algorithm stores the current state as an MPS at all times, which may be periodically compressed to improve efficiency. Figure (e) depicts the algorithm at completion: the measurements of all $n$ of the qudits have been simulated.
• Figure 2: Iteration of SEBD. In (a), we begin with an MPS describing the current state $\rho_j$. In (b), the MPS is compressed via truncation of small Schmidt values. This will generally decrease the bond dimension of the MPS, depicted by the thin black line rather than thick purple line. In (c), qudits acted on by $V_j$ that are not already incorporated into the current state are added to the MPS (increasing the physical bond dimension of the MPS) and initialized in $\ket{0}$ states. In (d), the unitary gates associated with $V_j$ are applied. Figure (e) depicts the MPS after the application of $V_j$; the thick purple lines schematically illustrate the fact that the bond dimension may increase in this step. In (f), the measurement of column j is performed, and the outcome 01101 is obtained. Subsequently column j is projected onto 01101, removing the physical legs associated with these sites from the MPS. The resulting state is $\rho_{j+1}$.
• Figure 3: Patching. Pink represents marginals of the output distribution that have been approximately sampled, while white represents unsampled regions. In $(a)$, the algorithm has sampled from disconnected patches. Figure $(b)$ depicts how the algorithm transitions from configuration $(a)$ to $(c)$. Namely, the algorithm generates a sample from the conditional distribution on $A$, conditioned on the configuration of region $B$. Similarly, figure $(d)$ depicts how the "holes" of configuration $(c)$ are filled in. The end result is shown in $(e)$, an approximate sample from the global distribution on the full lattice.
• Figure 4: Extended brickwork architecture with $n$ qubits. Here, circles represent qubits initialized in the state $\ket{0}^{\otimes n}$, blue lines represent the first layer of gates to act, orange lines represent the second layer, and black lines represent the third and final layer. All gates are chosen Haar-randomly. We let Brickwork$(L,r,v)$ denote the corresponding random circuit with circuit layout depicted in the figure above with vertical sidelength $L$, "extension parameter" $2r$ (which gives the distance between vertical gates acting on adjacent pairs of rows), and number of pairs of columns of vertical gates $v$. In the above example, $r=7$ and $v=4$. The standard brickwork architecture corresponds to $r=1$. Note that $n = \Theta(L r v)$.
• Figure 5: Illustration of the state after the qubits of columns $i,i+1,\dots,j$ have been measured, but before gates in the lightcone of registers $A$ and $L$ have been performed. In each row $i$, we are left with a post-measurement bipartite state $\ket{\phi_i}_{L_i R_i}$ depicted by a wavy line. The expected entanglement entropy $S(L_i)_{\phi_i}$ decays exponentially in $r$. The final state of interest $\ket{\psi'}$ is obtained by applying local unitaries to the qubits in the dashed red box before measuring all of these qubits in the computational basis, inducing the final state $\ket{\psi'}$ on $R = R_1 \cup \cdots \cup R_L$. By concavity of the von Neumann entropy, the expected entanglement entropy of $\ket{\psi'}$ across the cut defined by the dotted blue box is upper bounded by the entanglement entropy across this cut before the unitaries and measurements in the dashed red box are performed.

Theorems & Definitions (41)

• Definition 1: Architecture
• Definition 2: Uniformity
• Definition 3: Quasi-$k$ entropy
• Lemma 1
• Corollary 1
• Corollary 2
• Lemma 2
• Corollary 3
• proof
• Corollary 4