Paul Webster

Email: pwebster2105@gmail.com

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I study the theory of quantum computing. Specifically, my research is on developing and understanding approaches for fault-tolerant implementations of logic gates on quantum information. My work has primarily focussed on topological codes, but I have also become increasingly interested in more general quantum error-correcting codes.

I submitted my PhD thesis, supervised by Stephen Bartlett, in December 2020, and subsequently worked as a postdoctoral researcher in the Quantum Science Group at The University of Sydney.

Please feel welcome to email me if you are interested in my work.

More details on my completed research projects and publications follow.

Partitioning Qubits in Hypergraph Product Codes to Implement Logical Gates

Armanda O. Quintavalle, Paul Webster and Michael Vasmer
Partitioning qubits in hypergraph product codes to implement logical gates
Preprint: arXiv:2204.10812

Abstract:

The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.

Universal Fault-Tolerant Quantum Computing with Stabiliser Codes

Primary Publication:

Paul Webster, Michael Vasmer, Thomas R. Scruby and Stephen D. Bartlett
Universal Fault-Tolerant Quantum Computing with Stabiliser Codes
Published: Physical Review Research, 4, 013092, 2022
Preprint: arXiv:2012.05260

Abstract:

The quantum logic gates used in the design of a quantum computer should be both universal, meaning arbitrary quantum computations can be performed, and fault-tolerant, meaning the gates keep errors from cascading out of control. A number of no-go theorems constrain the ways in which a set of fault-tolerant logic gates can be universal. These theorems are very restrictive, and conventional wisdom holds that a universal fault-tolerant logic gate set cannot be implemented natively, requiring us to use costly distillation procedures for quantum computation. Here, we present a general framework for universal fault-tolerant logic with stabilizer codes, together with a no-go theorem that reveals the very broad conditions constraining such gate sets. Our theorem applies to a wide range of stabilizer code families, including concatenated codes and conventional topological stabilizer codes such as the surface code. The broad applicability of our no-go theorem provides a new perspective on how the constraints on universal fault-tolerant gate sets can be overcome. In particular, we show how nonunitary implementations of logic gates provide a general approach to circumvent the no-go theorem, and we present a rich landscape of constructions for logic gate sets that are both universal and fault-tolerant. That is, rather than restricting what is possible, our no-go theorem provides a signpost to guide us to new, efficient architectures for fault-tolerant quantum computing.

Conference Presentations:
TQC 2021 (Online, July 2021): Contributed Talk (video, slides)
QIP 2021 (Online, February 2021): Poster (file)

Secondary Publication:
Thomas R. Scruby, Dan E. Browne, Paul Webster and Michael Vasmer
Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code
Published: Quantum, 6, 721, 2022
Preprint: arXiv:2012.08536

Abstract:

We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) CCZ between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which CCZ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic X and measurement errors and find evidence for a threshold pc∼0.1% in all three codes. We expect that this threshold could be improved by optimisation of the decoder.

Fault-Tolerant Quantum Gates with Defects in Topological Stabilizer Codes

Paul Webster and Stephen D. Bartlett
Fault-tolerant quantum gates with defects in topological stabilizer codes
Published: Physical Review A, 102, 022403, 2020 (Editors' Suggestion)
Preprint: arXiv:1906.01045
Abstract:

Braiding defects in topological stabilizer codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realized by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher-dimensional defect-braiding schemes have the potential to play an important role in realizing fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher-dimensional codes, specifically, in self-dual surface codes; the three-dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic-state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

Conference Presentations:
TQC 2020 (Online, June 2020): Contributed Talk (video, slides)
QEC 2019 (London, July-August 2019): Poster (file)



Locality-Preserving Logical Operators in Topological Stabilizer Codes

Paul Webster and Stephen D. Bartlett
Locality-preserving logical operators in topological stabilizer codes
Published: Physical Review A, 97, 012330, 2018
Preprint: arXiv:1709.00020
Abstract:

Locality-preserving logical operators in topological codes are naturally fault tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabilizer codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including color codes and toric codes, as well as more general codes such as Abelian quantum double models and codes with fermionic excitations in more than two dimensions.

Conference Presentations:
TQC 2018 (Sydney, July 2018): Contributed Talk (slides)
AIP 2018 (Perth, December 2018): Contributed Talk
AQIS 2017 (Singapore, September 2017): Contributed Talk
QIP 2017 (Seattle, January 2017): Poster (file)



Reducing the Overhead for Quantum Computation when Noise is Biased

Paul Webster, Stephen D. Bartlett and David Poulin
Reducing the overhead for quantum computation when noise is biased
Published: Physical Review A, 92, 062309, 2015
Preprint: arXiv:1509.05032
Abstract:

We analyze a model for fault-tolerant quantum computation with low overhead suitable for situations where the noise is biased. The basis for this scheme is a gadget for the fault-tolerant preparation of magic states that enable universal fault-tolerant quantum computation using only Clifford gates that preserve the noise bias. We analyze the distillation of| T〉-type magic states using this gadget at the physical level, followed by concatenation with the 15-qubit quantum Reed-Muller code, and comparing our results with standard constructions. In the regime where the noise bias (rate of Pauli Z errors relative to other single-qubit errors) is greater than a factor of 10, our scheme has lower overhead across a broad range of relevant noise rates.