Efficient classical simulation of noisy quantum computation

TL;DR

This work analyzes the boundary between classical simulability and quantum advantage in noisy quantum circuits by developing a tensor-network framework to study time dynamics and Fourier spectra. It proves that at a constant per-gate noise level $ε$, generic noisy universal circuits are classically simulatable in time polynomial in the circuit size, even when a Clifford subset is noiseless, and it provides a polynomial-time algorithm for many such circuits when some gates remain noiseless. The key results are two theorems: (i) with full noise and an anti-concentration condition, most outputs are nearly uniform with additive error $δ=ce^{-εd}$; (ii) with partial noiseless Clifford regions, one can approximate outputs in time $ ext{poly}(n)ig(8mig)^l$ with $l oughly rac{ ext{ln}(cδ^{-1}(1+rac{1}{ oot 2ar{η}}))}{2ε}$. The paper introduces a separable Fourier-spectrum tensor-network approach that elucidates the subtle boundary between classical simulatability, quantum supremacy, and fault-tolerant computing, with potential applicability to broader open quantum systems.

Abstract

Understanding the boundary between classical simulatability and the power of quantum computation is a fascinating topic. Direct simulation of noisy quantum computation requires solving an open quantum many-body system, which is very costly. Here, we develop a tensor network formalism to simulate the time-dynamics and the Fourier spectrum of noisy quantum circuits. We prove that under general conditions most of the quantum circuits at any constant level of noise per gate can be efficiently simulated classically with the cost increasing only polynomially with the size of the circuits. The result holds even if we have perfect noiseless quantum gates for some subsets of operations, such as all the gates in the Clifford group. This surprising result reveals the subtle relations between classical simulatability, quantum supremacy, and fault-tolerant quantum computation. The developed simulation tools may also be useful for solving other open quantum many-body systems.

Figures (6)

• Figure 1: Two types of ensembles of noisy quantum circuits.a, All the gates represented by blue boxes are chosen uniformly from the four possible Pauli matrices $X^{a}Z^{b}$ with $a,b=0,1$ (including the identity matrix with $a=b=0$). The gates represented in white could be any gates in the universal gate set. The red circles represent the channel of noise that is specified in the main text. The depth $d$ of this circuit is defined as the smallest number of blue boxes along any horizontal lines. b, The grey color box represents any circuits composed by noiseless Clifford gates. The white boxes represents non-Clifford gates, which are normally chosen as the single-bit $\pi/8$-gate for universal quantum computation. The non-Clifford gates are subject to the noise channel represented by the red boxes.
• Figure 2: Tensor network representation of the noise channel. a, Illustration of a tensor network, where the shared edges represent summation over the corresponding indices ($i,j,k,l$) and the remaining edges represent free indices ($p,q,r$). b, Basic tensors and notations used in our construction: rank-$2$ tensors $M, A, B$ with two indices, which are actually matrices (notice that to match the convention of drawing quantum circuits, the left/right index in the horizontal line corresponds respectively to the right/left index for the matrix; rank-$3$ identity tensor which constraints all the indices to be equal; rank-$1$ tensors with one index representing quantum states; a unitary transformation on a density matrix. c, Basic transformations of a tensor network that are frequently used in our construction, where $X, Z$ are Pauli matrices and $|+\rangle=(|0\rangle+|1\rangle)/\sqrt 2$. d, Tensor network representation of a unitary transformation $Z^{z}$ (with a factor $1/2$, $z=0,1$) on the right side, which is equivalent to the network on the left side using the above basic transformation rules. e, Tensor network representation of a quantum noise channel $\mathcal{E}_\mathrm{z}(Z^y\rho Z^y)=(1-\epsilon)Z^y\rho Z^y+\epsilon Z^{y+1}\rho Z^{y+1}$ (with a factor $1/2$), which is a combination of a unitary transformation $Z^y$ and a dephasing noise channel $\mathcal{E}_\mathrm{z}$ with the error probability $\epsilon$ (the phase-flip rate). This representation can be derived from Fig. d by setting $z=y$ and $z=y+1$, respectively.
• Figure 3: Tensor network representation of the basic units of noisy quantum circuits.a, Representation of the Pauli matrix ensemble (the blue box) followed by the noise channel (the red dot). The blue box denotes the ensemble of gates $\{X^{y_2}Z^{y_1}\}$ with $y_1,y_2$ chosen randomly from $\{0,1\}$ with equal probability. The red dot denotes the noise channel $\mathcal{E}_\mathrm{x}(\mathcal{E}_\mathrm{z}(\rho))$, a combination of phase flip and bit-flip errors. The tensor network representation of this noise channel is derived from Fig. 2e by noticing that the additional bit flip channel is obtained through the Hadamard transformation $H$ with $HZH=X$ and $H\mathcal{E}_\mathrm{z}(Z^y\rho Z^y)H=\mathcal{E}_\mathrm{x}(X^y H\rho H X^y)$. We assume the noisy quantum circuit starts at an initial state denoted as $|0\rangle$ and ends with a measurement in the computational basis $|x\rangle$, and the contraction of the tensor network gives the joint probability distribution $q^\prime_{x,y_1,y_2}$. b, Representation of the noiseless Pauli matrix ensemble (the blue box only). Contraction of the tensor network in this case gives the joint probability distribution $q_{x,y_1,y_2}$ for the corresponding noiseless circuit.
• Figure 4: Tensor network representation of Fourier transformation of noisy quantum circuits. a, Tensor Representation of Fourier transformation on boolean variables $y_i$, which is equivalent to the Hadamard transformation on the corresponding indices. The inverse Fourier transformation is the same because of $H^{2}=I$. b, Representation of the Fourier transformation on noisy circuits. As $HXH=Z$ and $[(1-\epsilon )I+\epsilon Z]|s\rangle =(1-2\epsilon )^{s}|s\rangle$, we immediately get that the Fourier spectrum of the noisy circuit is related to the spectrum of the noiseless circuit by a simple multiplication factor $(1-2\epsilon )^{s}$.
• Figure 5: Separability for the Tensor network representation of the Fourier spectrum of quantum circuits.a, The pivotal property of separability for the Fourier spectrum, which shows that the Fourier transformation of the Pauli ensemble (the basic unit denoted by the blue box in Fig. 3b) on the left side can be written as a product of two separable tensor networks on the right side, where $\sigma_{ab}$ denotes the matrix $X^{a}Z^{b}$. This separability is derived in Figs. b-d through the diagrammatic method. The separability is the key to simplification of the calculation of the Fourier spectrum for ensembles of large quantum circuits. b, Step 1 to derive the separability which uses the equalities in Fig. \ref{['fig:MT_tensors']}c. c, Step 2 to derive the separability in the case with $a=b=0$. d, Step 3 to derive the separability for the general case with arbitrary $a,b$. We have used the result of Fig. 5b and the tensor property in Fig. 2c.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2