Efficient simulation of logical magic state preparation protocols

TL;DR

This work presents a scalable framework for simulating logical magic state preparation (MSP) protocols under circuit-level Pauli noise by showing that circuit-level errors propagate to a Clifford error at the end of the circuit. Central to the approach is the Pauli-Square-Root Clifford (PSC) structure, which enables a canonical decomposition of MSP circuits into end-of-circuit Clifford propagations and a subsequent stabilizer-based fidelity estimation. The authors develop phase-insensitive (Pauli-rank) and phase-sensitive (stabilizer-rank) fidelity estimation methods, and validate the technique with a proof-of-principle MSP on a Steane code, achieving accurate results with substantial speedups over full state-vector simulations. The framework applies to a broad class of MSP protocols, including code-switching, magic-state cultivation, and distillation, and yields polynomial-time scaling in the number of qubits and the target magic state's non-stabilizerness. This work enables practical benchmarking and analysis of large-scale MSP protocols without resorting to exhaustive simulations.

Abstract

Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.

Figures (15)

• Figure 1: Schematic representation of the simulation technique for a magic state preparation (MSP) protocol. (a) This is the general form of the MSP protocol considered in this paper. Here $U_1,\ldots, U_\ell$ are layers of unitary gates that comprise the protocol. Circuit-level Pauli noise is denoted $P_1,\ldots, P_{\ell}$, $P_I$, and $P_M$, representing the errors associated with $U_1,\ldots, U_\ell$, initialization, and measurement, respectively. Upon measuring the ancilla register (denoted by $|a\rangle$), one either post-selects or error-corrects the logical magic state. (b) For our circuit family, all circuit-level Pauli errors can be propagated to an end-of-circuit Clifford error $C_\mathrm{prop}$. (c) Because a noiseless circuit would have prepared a noiseless magic state, the original noisy protocol circuit can be recast as a Clifford error $C_\mathrm{prop}$ acting on a noiseless logical magic state $|\bar{\phi}\rangle \otimes |a\rangle= \Pi_i U_i(|\bar{\psi}\rangle \otimes |a\rangle)$. (d) To enable efficient simulation, the logical magic state is decomposed into a linear combination of stabilizer states $|\bar{s}_i\rangle$. The fidelity of the output noisy logical magic state can be obtained by simulating an ensemble of logical stabilizer states subjected to stochastic Clifford errors and measurements in the Pauli basis; see Section \ref{['sec:sim_technique']} for more details.
• Figure 2: Pictorial representation of the $\mathbf{[[4,2,2]]}$ code. Each circle corresponds to a physical qubit with their respective labels. The stabilizer group generators (solid lines) and logical Pauli-operators (dashed lines) are highlighted.
• Figure 3: A toy example protocol for preparing an encoded magic state $|\bar{H}\rangle |\bar{H}\rangle$ in the $[[4,2,2]]$ code. The logical magic state is prepared, followed by two rounds of logical Clifford and stabilizer measurements.
• Figure 4: Example of $X$-error propagation in $\left| \bar{H} \right\rangle\left| \bar{H} \right\rangle$-state measurement protocol on $[[4,2,2]]$ code. Left: Injection of a Pauli $X$-error. Right: The $X$-error propagates to a Clifford error at the end of the circuit.
• Figure 5: Non-FT standard measurement gadget. Measurement of a transversal logical PSC $\bar{U}=\bigotimes_{i=1}^\ell V_i$, for PSCs $V_i$, is followed by single-ancilla measurements of Pauli stabilizer generators $\{g_1,...,g_{m}\}$ where $g_i=\bigotimes_{j=1}^{t}P_j^{(i)}$ for single-qubit Paulis $P_j^{(i)}$.

Theorems & Definitions (23)

• Definition 1
• Lemma 1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 1
• proof
• Definition 2: Tranversal PSC gate
• Theorem 2
• Corollary 1