Extractors: QLDPC Architectures for Efficient Pauli-Based Computation

TL;DR

This work introduces extractor systems to enable universal, fault-tolerant Pauli-based computation on quantum LDPC memories. It defines extractor-augmented computation (EAC) blocks and fixed-connectivity architectures connected by bridges/adapters, allowing fault-tolerant measurement of any logical Pauli operator in one logical cycle and scalable parallel computation when supplied with high-fidelity magic states. By reducing logical operations to Pauli measurements and magic-state consumption, the approach achieves universal computation with bounded extractor overhead ($O(n(\log n)^3)$ qubits) and a computable depth bound for Clifford+$T$ circuits, along with a flexible design space for hardware constraints. The paper also develops a comprehensive surgery toolkit for constructing measurement graphs, discusses practical considerations, and outlines uniform vs hybrid architectures, highlighting open questions in optimization, decoding, and realistic resource estimates for specific codes and hardware platforms.

Abstract

In pursuit of large-scale fault-tolerant quantum computation, quantum low-density parity-check (LDPC) codes have been established as promising candidates for low-overhead memory when compared to conventional approaches based on surface codes. Performing fault-tolerant logical computation on QLDPC memory, however, has been a long standing challenge in theory and in practice. In this work, we propose a new primitive, which we call an $\textit{extractor system}$, that can augment any QLDPC memory into a computational block well-suited for Pauli-based computation. In particular, any logical Pauli operator supported on the memory can be fault-tolerantly measured in one logical cycle, consisting of $O(d)$ physical syndrome measurement cycles, without rearranging qubit connectivity. We further propose a fixed-connectivity, LDPC architecture built by connecting many extractor-augmented computational (EAC) blocks with bridge systems. When combined with any user-defined source of high fidelity $|T\rangle$ states, our architecture can implement universal quantum circuits via parallel logical measurements, such that all single-block Clifford gates are compiled away. The size of an extractor on an $n$ qubit code is $\tilde{O}(n)$, where the precise overhead has immense room for practical optimizations.

Figures (11)

• Figure 1: High level depiction of an extractor architecture paired with a magic state factory. (a) Extractor-augmented computational (EAC) blocks $\mathcal{Q} \leftrightarrows \mathcal{X}$ connected by bridges $\mathcal{B}$. In our architecture, these EAC blocks store and operate on logical information via logical Pauli measurements. (b) A magic state factory (colored gears) supplying high-fidelity $\ket{T}$ magic states to individual EAC blocks. The output magic states may be stored in local caches, which are connected via adapters $\mathcal{A}$ to the EAC blocks. If the caches are themselves high-rate QLDPC memories, they can also be equipped with extractors (not drawn) to facilitate the storage and consumption of magic states.
• Figure 2: Example of circuit compilation for our extractor architecture. The colored components are the ones that we compile and execute on EAC blocks, and the uncolored ones are compiled away. (a) A circuit composed of Pauli $\pi/4$ rotations (which are Clifford gates), $Z_{\pi/8}$ rotations, and $Z$ measurements at the end. Qubits are grouped into EAC blocks of size 3. (b) Exchanging all $\pi/8$ rotations and cross-block $\pi/4$ rotations to the beginning of the circuit, and all in-block Clifford gates to the end. (c) Absorbing all in-block Clifford gates into the final measurements. The remaining Pauli rotations will be implemented by Pauli measurements, as depicted in Figure \ref{['fig:compilation']} of Methods.
• Figure 3: Logical measurement of an operator $\mathcal{L}$ using a measurement graph $G$, depicted with scalable Tanner graphs. Here, groups of circles denote qubits, and groups of squares denote checks. Lines between checks and qubits are labelled by sympletic matrices, denoting the Pauli actions the checks have on the qubits. (a) Tanner graph of the code $\mathcal{Q}$. The qubits on the right labelled $L$ are qubits in support of $\mathcal{L}$, and checks on the right labelled $\mathcal{S}_\mathcal{L}$ are checks where $K(\mathcal{S}, \mathcal{L})\ne\varnothing$ (see Definition \ref{['def:graph_and_code']}, checks \ref{['stab:modified_code_checks']}). Unlabelled qubits on the left are the remaining qubits in $Q\setminus L$, unlabelled checks on the left are remaining checks with $K(\mathcal{S}, \mathcal{L}) = \varnothing$. Checks act on qubits as specified by the sympletic stabilizer matrix $[S_X\vert S_Z]$ of $\mathcal{Q}$. (b) Ancilla system specified by the measurement graph $G$. Every edge in $G$ is an ancilla qubit. Vertex checks $V$ act on edge qubits by $Z$ with incidence matrix $M_G$, cycle checks from basis $R$ act on edge qubits by $X$ with incidence matrix $M_R$. (c) The code and graph systems are coupled by check deformation. The vertex checks $V$ act on qubits in $L$ as specified by the port function $f$ and the operator $\mathcal{L}$. The code checks $\mathcal{S}_\mathcal{L}$ act on edge qubits that form path matchings by $X$ as specified by Definition \ref{['def:graph_and_code']}, checks \ref{['stab:modified_code_checks']}.
• Figure 4: Depicting of thickening, decongestion, and cellulation. (a) A generic graph $G_1$. (b)$G_1$ thickened by a line graph $J_3$, $G = G_1\square J_3$. (c) A cycle basis in $G$, where the blue cycles are a cycle basis of $G_1$ spread into distinct levels and therefore do not overlap, and the red cycle is one of the new cycles created by thickening (Fact \ref{['fact:thickened_cycles']}). (d) Cellulating cycles into triangles. We did not cellulate the red cycle(s) as they always have weight 4.
• Figure 5: Depiction of different components of an Extractor-Augmented Computation (EAC) block. In all panels, circles denote data qubits, sqaures denote check qubits, and lines denote connections between check and data qubits. The color of a line denote the Pauli action of the check qubit on the data qubit: red for $Z$, blue for $X$. (a) A generic quantum CSS code $\mathcal{Q}$ with data and check qubits. We use dotted lines to indicate that these connections came with the code. The lines are uncolored as their Pauli actions are unspecified. (b) The first level of an extractor $\mathcal{X}$ (light green) has one check qubit per data qubit in $\mathcal{Q}$. They are connected 1-to-1. The lines are uncolored as their Pauli actions are unspecified. (c) For every stabilizer $S$ of $\mathcal{Q}$, we add a cycle $C$ of edges among the vertex checks that are connected to the qubits in support of $S$. Each edge is a data qubit in $\mathcal{X}$. The vertex checks act on edge qubits by $Z$. The stabilizer $S$ are extended to act on the edge qubits by $X$. Together, panels b, c depict all the coupling edges $\leftrightarrows$ between $\mathcal{Q}$ and $\mathcal{X}$. (d) The base graph (first level) of an extractor $X_1$ is a constant degree expander graph. In practice, due to the underlying code structure, one should consider co-designing the extractor graph with the code, see Section \ref{['sec:extractor_practical']}. (e) An extractor may have multiple levels due to thickening (Definition \ref{['def:thicken']}). (f) Create cycle checks for a basis of cycles of $X$. We depict two types of cycles here: the vertical cycle between levels of $X$ comes from thickening (Fact \ref{['fact:thickened_cycles']}), and the horizontal cycles on each level come from $X_1$. Cycle checks act on edge qubits by $X$. The circuit with dashed boundary denote a qubit that came from cellulation (Definition \ref{['def:cellulation']}).

Theorems & Definitions (67)

• Remark 1: Equivalent perspectives on QLDPC surgery
• Definition 2: Measurement Graphs and Codes
• Remark 3
• Definition 4: Cycle Basis and Congestion
• Definition 5: Path Matching
• Remark 6
• Theorem 7: Graph Desiderata williamson2024lowswaroop2024universal
• Definition 8: Cheeger Constant and Relative Expansion
• Remark 9
• Definition 10: Measurement Protocol williamson2024low