Classical Simulability of Quantum Circuits with Shallow Magic Depth

TL;DR

This work investigates the classical simulability of quantum circuits that concentrate non-Clifford gates into a shallow magic depth, focusing on three tasks: amplitude estimation, sampling, and Pauli observable evaluation. It shows a nuanced landscape: amplitudes and sampling at multiplicative error for TD1 circuits are GapP-hard under plausible assumptions, while Pauli observables are efficiently computable in TD1 due to CH$_3$ structure; adding a single extra magic layer or switching to $T^{1/2}$ gates triggers a sharp transition to GapP-hardness for Pauli observables. When additive error $1/ ext{poly}(n)$ is allowed, a polynomial-time classical algorithm exists for amplitudes, Pauli observables, and marginal sampling from $ ext{log}(n)$ qubits for diagonal magic-depth-one circuits, illustrating a regime where classical simulation is feasible. A path-integral technique is presented for circuits with more than one magic layer, offering sub-exponential scaling in the number of magic layers. Overall, the results highlight the critical role of magic depth in determining classical simulability and offer practical algorithms for specific regimes, thereby delineating the boundary between classical simulability and quantum advantage in shallow-magic circuits.

Abstract

Quantum magic is a necessary resource for quantum computers to be not efficiently simulable by classical computers. Previous results have linked the amount of quantum magic, characterized by the number of $T$ gates or stabilizer rank, to classical simulability. However, the effect of the distribution of quantum magic on the hardness of simulating a quantum circuit remains open. In this work, we investigate the classical simulability of quantum circuits with alternating Clifford and $T$ layers across three tasks: amplitude estimation, sampling, and evaluating Pauli observables. In the case where all $T$ gates are distributed in a single layer, performing amplitude estimation and sampling to multiplicative error are already classically intractable under reasonable assumptions, but Pauli observables are easy to evaluate. Surprisingly, with the addition of just one $T$ gate layer or merely replacing all $T$ gates with $T^{\frac{1}{2}}$, the Pauli evaluation task reveals a sharp complexity transition from P to GapP-complete. Nevertheless, when the precision requirement is relaxed to 1/poly($n$) additive error, we are able to give a polynomial time classical algorithm to compute amplitudes, Pauli observable, and sampling from $\log(n)$ sized marginal distribution for any magic-depth-one circuit that is decomposable into a product of diagonal gates. Our research provides new techniques to simulate highly magical circuits while shedding light on their complexity and their significant dependence on the magic depth.

Figures (8)

• Figure 1: Circuits considered in this paper. Clifford gates are marked black and non-Clifford gates are marked in magenta color. (a) The task of computing amplitude in magic-depth-one circuits. $D_i$ are magic gates acting on $O(1)$ qubits, while $U_{c,l}$ and $U_{c,r}$ are Clifford unitaries sandwiching the magic layer. (b) The task of computing Pauli observable in magic-depth-one circuits. Note that we remove $U_{c,r}$ and replace $P$ with $U_{c,r}^\dag P U_{c,r}$. (c) An example of the degree-three IQP circuits. The magic gates $CCZ$ are in magenta color.
• Figure 2: The parallelization trick. One introduces a set of ancilla (shown in blue) for each diagonal gate $D_i$ and copy the bitstring values to the ancilla. The phase gates are then applied simultaneously. Ancilla are cleaned in the end.
• Figure 3: Decomposition of the $CCZ$ gate. (a) Decomposing $CCZ$ gate into $CNOT$ gates and $T^{\pm 1}$ gates. (b) Compiling the circuit in (a) into one layer of $T^{\pm 1}$ gates.
• Figure 4: Computing Pauli observable and the Hadamard test. (a) Evolving Pauli operators with gates from $\mathcal{CH}_3$ turn them into Hermitian Clifford operator. (b) The Hadamard test reduces computing ampiltudes to computing Pauli observable. (c) The Hadamard test that compute the amplitude of the degree-three IQP circuit shown in Fig. \ref{['fig:td1_setup_iqp']}(c). $CCZ$ gates are in magenta and $CCCZ$ gates are in yellow color.
• Figure 5: Synthesis of the $C^l Z$ gate Synthesizing $C^lZ$ gates with two layers of diagonal gates from $\mathcal{D}_{m+1}$, where $l \le 2m$. The magenta gates are in $\mathcal{D}_{m+1}$, while the gate $X^{-\frac{1}{2}}$ is a Clifford gate. $X^{-\frac{1}{2}}$ prevents the two magic layers from being parallelized into one layer.

Theorems & Definitions (36)

• Definition 2.1
• Proposition 2.1
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.2
• Lemma 2.2
• proof
• Lemma 2.3
• proof