arXiv subjects

Dominic J. Williamson

Publications and source records attributed to Dominic J. Williamson.

At least 37 records · Page 2Linked to original sources

arXiv2026

Fast magic state preparation by gauging higher-form transversal gates in parallel

Magic states are a foundational resource for universal quantum computation. To survive in a realistic noisy environment, magic states must be prepared fault-tolerantly and protected by a quantum error-correcting code. The recent discovery of highly efficient quantum low-density parity-check codes, together with efficient logic gates, lays the groundwork for low-overhead fault-tolerant quantum computation. This motivates the search for fast and parallel protocols for logical magic state preparation to enable universal quantum computation. Here, we introduce a fast code surgery procedure that performs a fault-tolerant measurement of many transversal logic gates in parallel. This is achieved by performing a generalized gauging measurement on a quantum code that supports a higher-form transversal gate. The time overhead of our procedure is constant, and the qubit overhead is linear. The procedure inherits fault-tolerance properties from the base code and the structure of the higher-form transversal gate. When applied to codes that support higher-form Clifford gates our procedure achieves fast and fault-tolerant preparation of many magic states in parallel. This motivates the search for good quantum low-density parity-check codes that support higher-form Clifford gates.

quant-ph↗

arXiv2026

Parallel Logical Measurements via Quantum Code Surgery

Quantum code surgery is a flexible and low overhead technique for performing logical measurements on quantum error-correcting codes, which generalises lattice surgery. In this work, we present a code surgery scheme, applicable to any qubit stabiliser low-density parity check (LDPC) code, that fault-tolerantly measures many logical Pauli operators in parallel. For a collection of logically disjoint Pauli product measurements supported on $t$ logical qubits, our scheme uses $O\big(t ω(\log t + \log^3ω)\big)$ ancilla qubits, where $ω\geq d$ is the maximum weight of the single logical Pauli representatives involved in the measurements, and $d$ is the code distance. This is all done in time $O(d)$ independent of $t$. Our proposed scheme preserves both the LDPC property and the fault-distance of the original code, without requiring ancillary logical codeblocks which may be costly to prepare. This addresses a shortcoming of several recently introduced surgery schemes which can only be applied to measure a limited number of logical operators in parallel if they overlap on data qubits.

quant-ph↗

arXiv2026

There and Back Again: A Gauging Nexus between Topological and Fracton Phases

Coupled layer constructions are a valuable tool for capturing the universal properties of certain interacting quantum phases of matter in terms of the simpler data that characterizes the underlying layers. In the study of fracton phases, the X-Cube model in 3+1D can be realized via such a construction by starting with a stack of 2+1D Toric Codes and turning on a coupling which condenses a composite "particle-string" object. In a recent work [Phys. Rev. B 112, 125124 (2025)], we have demonstrated that in fact, the particle-string can be viewed as a symmetry defect of a topological 1-form symmetry. In this paper, we study the result of gauging this symmetry in depth. We unveil a rich gauging web relating the X-Cube model to symmetry protected topological (SPT) phases protected by a mix of subsystem and higher-form symmetries, subsystem symmetry fractionalization in the 3+1D Toric Code, and non-trivial extensions of topological symmetries by subsystem symmetries. Our work emphasizes the importance of topological symmetries in non-topological, geometric phases of matter.

cond-mat.str-el↗

arXiv2025

Dynamical quantum codes and logic gates on a lattice with sparse connectivity

We introduce several dynamical schemes that take advantage of mid-circuit measurement and nearest-neighbor gates on a lattice with maximum vertex degree three to implement topological codes and perform logic gates between them. We first review examples of Floquet codes and their implementation with nearest-neighbor gates and ancillary qubits. Next, we describe implementations of these Floquet codes that make use of the ancillary qubits to reset all qubits every measurement cycle. We then show how switching the role of data and ancilla qubits allows a pair of Floquet codes to be implemented simultaneously. We describe how to perform a logical Clifford gate to entangle a pair of Floquet codes that are implemented in this way. Finally, we show how switching between the color code and a pair of Floquet codes, via a depth-two circuit followed by mid-circuit measurement, can be used to perform syndrome extraction for the color code.

quant-ph↗

arXiv2025

Extractors: QLDPC Architectures for Efficient Pauli-Based Computation

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.

quant-ph↗

arXiv2025

Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery

Quantum code surgery is a promising technique to perform fault-tolerant computation on quantum low-density parity-check codes. Recent developments have significantly reduced the space overhead of surgery. However, generic surgery operations still require $O(d)$ rounds of repeated syndrome extraction to be made fault-tolerant. In this work, we focus on reducing the time overhead of surgery. We first present a general set of conditions that ensure fault-tolerant surgery operations can be performed with constant time overhead. This fast surgery necessarily makes use of an auxiliary complex described by a hypergraph rather than a graph. We then introduce a concrete scheme called block reading, which performs transversal surgery across multiple code blocks. We further investigate surgery operations with intermediate time overhead, between $O(1)$ and $O(d)$, which apply to quantum locally testable codes. Finally, we establish a circuit equivalence between homomorphic measurement and hypergraph surgery and derive bounds on the time overhead of generic logical measurement schemes. Overall, our results demonstrate that reducing the time cost of code surgery is not reliant on the quantum memory being single-shot. Instead it is chiefly the connectivity between a code and its measurement ancilla system that determines the achievable measurement time overhead.

quant-ph↗

arXiv2025

Partial Self-Correction in Layer Codes

The storage of large-scale quantum information at finite temperature requires an autonomous and reliable quantum hard drive, also known as a self-correcting quantum memory. It is a long-standing open problem to find a self-correcting quantum memory in three dimensions. The recently introduced Layer Codes achieve the best possible scaling of code parameters and logical energy barrier in three dimensions, these are tantalizing features for the purposes of self-correction. In this work we show that a family of Layer Codes, based on good Quantum Tanner Codes, exhibit partial self-correction. Their memory time grows exponentially with linear system size, up to a length scale that is exponential in the inverse temperature. At this length scale, the memory time scales as a double exponential of inverse temperature. To establish this result we introduce a concatenated matching decoder that combines three rounds of parallelized minimum-weight perfect-matching with a decoder for good Quantum Tanner Codes. We show that our decoder corrects errors up to a constant fraction of the energy barrier, and a constant fraction of the code distance, for a family of Layer Codes. Our results position Layer Codes as the leading candidate for a partially self-correcting memory in three dimensions. While they fall short of achieving strict self-correction in the thermodynamic limit, our work highlights the potential of these local codes in three dimensions, with fast distance and logical qubit growth, fast decoders, and a long memory time over a wide range of parameters.

quant-ph↗

arXiv2025

Thresholds for post-selected quantum error correction from statistical mechanics

We identify regimes where post-selection can be used scalably in quantum error correction (QEC) to improve performance. We use statistical mechanical models to analytically quantify the performance and thresholds of post-selected QEC, with a focus on the surface code. Based on the non-equilibrium magnetization of these models, we identify a simple heuristic technique for post-selection that does not require a decoder. Along with performance gains, this heuristic allows us to derive analytic expressions for post-selected conditional logical thresholds and abort thresholds of surface codes. We find that such post-selected QEC is characterised by four distinct thermodynamic phases, and detail the implications of this phase space for practical, scalable quantum computation.

quant-ph↗

arXiv2025

Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits

The classification of topological phases of matter is a fundamental challenge in quantum many-body physics, with applications to quantum technology. Recently, this classification has been extended to the setting of Adaptive Finite-Depth Local Unitary (AFDLU) circuits which allow global classical communication. In this setting, the trivial phase is the collection of all topological states that can be prepared via AFDLU. Here, we propose a complete classification of the trivial phase by showing how to prepare all solvable anyon theories that admit a gapped boundary via AFDLU, extending recent results on solvable groups. Our construction includes non-Abelian anyons with irrational quantum dimensions, such as Ising anyons, and more general acyclic anyons. Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase described by a solvable anyon theory with gapped boundary. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons and symmetry twist defects in solvable anyon theories. We apply our procedure to the quantum double of the group $S_3$ and to several examples that are beyond solvable groups, including the doubled Ising theory, the $\mathbb{Z}_3$ Tambara-Yamagami string-net, and doubled $SU(2)_4$ anyons.

quant-ph↗

arXiv2025

String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation

We present a new perspective on the $p$-string condensation procedure for constructing 3+1D fracton phases by implementing this process via the gauging of higher-form symmetries. Specifically, we show that gauging a 1-form symmetry in 3+1D that is generated by Abelian anyons in isotropic stacks of 2+1D topological orders naturally results in a 3+1D $p$-string condensed phase, providing a controlled non-perturbative construction that realizes fracton orders. This approach clarifies the symmetry principles underlying $p$-string condensation and generalizes the familiar connection between anyon condensation and one-form gauging in two spatial dimensions. We demonstrate this correspondence explicitly in both field theories and lattice models: in field theory, we derive the foliated field theory description of the $\mathbb{Z}_N$ X-Cube model by gauging a higher-form symmetry in stacks of 2+1D $\mathbb{Z}_N$ gauge theories; on the lattice, we show how gauging a diagonal 1-form symmetry in isotropic stacks of $G$-graded string-net models leads to string-membrane-nets hosting restricted mobility excitations. This perspective naturally generalizes to spatial dimensions $d \geq 2$ and provides a step towards building an algebraic theory of $p$-string condensation.

cond-mat.str-el↗

arXiv2025

Universal fault tolerant quantum computation in 2D without getting tied in knots

We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose codespace is stabilized by Clifford operators. We formulate a path integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus, establishing a new connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.

quant-ph↗

arXiv2025

Quantum cellular automata and categorical dualities of spin chains

Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators that respect the unit matrix-product operator. For conventional global symmetries, which are on-site representations of finite groups, this larger algebra is simply the tensor product of algebras associated to individual spins in the chain. We present a solution to the extension problem using the machinery of Doplicher-Haag-Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.

math-ph↗

arXiv2025

Subsystem Symmetry Fractionalization and Foliated Field Theory

Topological quantum matter exhibits a range of exotic phenomena when enriched by subdimensional symmetries. This includes new features beyond those that appear in the conventional setting of global symmetry enrichment. A recently discovered example is a type of subsystem symmetry fractionalization that occurs through a different mechanism to global symmetry fractionalization. In this work we extend the study of subsystem symmetry fractionalization through new examples derived from the general principle of embedding subsystem symmetry into higher-form symmetry. This leads to new types of symmetry fractionalization that are described by foliation dependent higher-form symmetries. This leads to field theories and lattice models that support previously unseen anomalous subsystem symmetry fractionalization. Our work expands the range of exotic topological physics that is enabled by subsystem symmetry in field theory and on the lattice.

cond-mat.str-el↗

arXiv2024

Braiding for the win: Harnessing braiding statistics in topological states to win quantum games

Nonlocal quantum games provide proof of principle that quantum resources can confer advantage at certain tasks. They also provide a compelling way to explore the computational utility of phases of matter on quantum hardware. In a recent manuscript [Hart et al., arXiv:2403.04829] we demonstrated that a toric code resource state conferred advantage at a certain nonlocal game, which remained robust to small deformations of the resource state. In this manuscript we demonstrate that this robust advantage is a generic property of resource states drawn from topological or fracton ordered phases of quantum matter. To this end, we illustrate how several other states from paradigmatic topological and fracton ordered phases can function as resources for suitably defined nonlocal games, notably the three-dimensional toric-code phase, the X-cube fracton phase, and the double-semion phase. The key in every case is to design a nonlocal game that harnesses the characteristic braiding processes of a quantum phase as a source of contextuality. We unify the strategies that take advantage of mutual statistics by relating the operators to be measured to order and disorder parameters of an underlying generalized symmetry-breaking phase transition. Finally, we massively generalize the family of games that admit perfect strategies when codewords of homological quantum error-correcting codes are used as resources.

quant-ph↗

arXiv2024

Layer Codes

The surface code is a two-dimensional topological code with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here, we introduce a construction that takes as input a stabilizer code and produces as output a three-dimensional topological code with related code parameters. The output codes have the special structure of being topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good low-density parity-check codes, the output is a three-dimensional topological code with optimal scaling code parameters and a polynomial energy barrier.

quant-ph↗

arXiv2024

Playing nonlocal games across a topological phase transition on a quantum computer

Many-body quantum games provide a natural perspective on phases of matter in quantum hardware, crisply relating the quantum correlations inherent in phases of matter to the securing of quantum advantage at a device-oriented task. In this paper we introduce a family of multiplayer quantum games for which topologically ordered phases of matter are a resource yielding quantum advantage. Unlike previous examples, quantum advantage persists away from the exactly solvable point and is robust to arbitrary local perturbations, irrespective of system size. We demonstrate this robustness experimentally on Quantinuum's H1-1 quantum computer by playing the game with a continuous family of randomly deformed toric code states that can be created with constant-depth circuits leveraging mid-circuit measurements and unitary feedback. We are thus able to tune through a topological phase transition - witnessed by the loss of robust quantum advantage - on currently available quantum hardware. This behavior is contrasted with an analogous family of deformed GHZ states, for which arbitrarily weak local perturbations destroy quantum advantage in the thermodynamic limit. Finally, we discuss a topological interpretation of the game, which leads to a natural generalization involving an arbitrary number of players.

quant-ph↗

arXiv2024

3-Fermion topological quantum computation

We present a scheme for universal topological quantum computation based on Clifford complete braiding and fusion of symmetry defects in the 3-Fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realisation of this computational scheme on the lattice using ground states of the Walker--Wang model for the 3-Fermion anyon theory with symmetry defects. The Walker--Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-Fermion anyon theory can be realized in a 2D subsystem code due to Bombín -- making contact with an alternative implementation of the 3-Fermion defect computation scheme via code deformations.

quant-ph↗

arXiv2023

Pauli topological subsystem codes from Abelian anyon theories

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories--this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the $\mathbb{Z}_4^{(1)}$ anyon theory with degenerate braiding relations and the chiral semion theory--both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.

quant-ph↗