Identity check problem for shallow quantum circuits

TL;DR

A classical algorithm approximating the distance to the identity within a factor $\alpha=D+1$ for shallow geometrically local $D$-dimensional circuits provided that the circuit is sufficiently close to the identity provided that the circuit is sufficiently close to the identity.

Abstract

Checking whether two quantum circuits are approximately equivalent is a common task in quantum computing. We consider a closely related identity check problem: given a quantum circuit $U$, one has to estimate the diamond-norm distance between $U$ and the identity channel. We present a classical algorithm approximating the distance to the identity within a factor $α=D+1$ for shallow geometrically local $D$-dimensional circuits provided that the circuit is sufficiently close to the identity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit depth and spatial dimension. We also show that the operator-norm distance to the identity $\|U-I\|$ can be efficiently approximated within a factor $α=5$ for shallow 1D circuits and, under a certain technical condition, within a factor $α=2D+3$ for shallow $D$-dimensional circuits. A numerical implementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100 qubits.

Figures (3)

• Figure 1: Eigenvalue polygon $P_U$ whose vertices are eigenvalues of $U$. The diamond-norm distance between $U$ and the identity channel is $\delta(U)=2\sqrt{1-r^2}$, where $r$ is the distance between $P_U$ and the origin aharonov1998quantum. If $P_U$ does not contain the origin then $\delta(U)$ coincides with the diameter of $P_U$. Otherwise, $\delta(U)=2$.
• Figure 2: Examples of reclusive partitions for $D=1,2$. Qubits are located at cells of a $D$-dimensional rectangular array. The array is partitioned into $D+1$ sets $A_1,\ldots,A_{D+1}$ such that each set $A_j$ is a disjoint union of $D$-dimensional cubes of linear size $L$ and the distance between any pair of cubes from the same set $A_j$ is at least $L/D$. Here $L=4$. Cubes located near the boundary of the array are truncated. The sets $A_1,A_2,A_3$ are highlighted in yellow, green, and blue.
• Figure 3: A comparison between the exact diamond-norm distance $\delta(U)$ (green dots) computed by a mapping to free fermions, an upper bound $\gamma$ computed by Algorithm 1 (blue dots) and the lower bound $\gamma/2$ (orange dots). Both bounds closely capture the exact distance between $U$ and $I$, demonstrating the scalability of our algorithm.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• proof : Proof of Lemma \ref{['lemma:comm1']}
• proof : Proof of Lemma \ref{['lemma:reduction']}
• Lemma 3: Reclusive Partitions woude2022geometry
• Lemma 4
• proof
• Definition 1
• Lemma 5: Additivity
• proof