Nikolas P. Breuckmann
@ucl.ac.uk
Department of Computer Science
University College London
RESEARCH INTERESTS
Quantum Information
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
Anurag Anshu, Nikolas P. Breuckmann, and Chinmay Nirkhe
ACM
The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that a particular family of constant-rate and linear-distance qLDPC codes correspond to NLTS local Hamiltonians, although we believe this to be true for all current constructions of good qLDPC codes.
Oscar Higgott and Nikolas P. Breuckmann
American Physical Society (APS)
In this work we study the single-shot performance of higher dimensional hypergraph product codes decoded using belief-propagation and ordered-statistics decoding [Panteleev and Kalachev, 2021]. We find that decoding data qubit and syndrome measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes. For the 3D toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for $Z$ errors, compared to the threshold of 2.90% previously found using a two-stage decoder~[Quintavalle et al., 2021]. For the 4D toric code, for which both $X$ and $Z$ error correction is single-shot, our results are consistent with a sustainable single-shot threshold of 4.3% which is even higher than the threshold of 2.93% for the 2D toric code for the same noise model but using $L$ rounds of stabiliser measurement. We also explore the performance of balanced product and 4D hypergraph product codes which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.
Benedikt Placke and Nikolas P. Breuckmann
American Physical Society (APS)
We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the euclidean case. We explain this anomaly by a careful re-derivation of the Kramers--Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature $T$ and the fraction of antiferromagnetic bonds $p$. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.
Anurag Anshu and Nikolas P. Breuckmann
AIP Publishing
The NLTS (No Low-Energy Trivial State) conjecture [M. H. Freedman and M. B. Hastings, Quantum Inf. Comput. 14, 144 (2014)] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity measured by the quantum circuit depth preparing the state). Here, we prove a weaker version called the combinatorial no low error trivial states (NLETS), where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms. This generalizes the prior NLETS results [L. Eldar and A. W. Harrow, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2017), pp. 427–438] and [Nirkhe et al., in 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Leibniz International Proceedings in Informatics (LIPIcs), edited by Chatzigiannakis et al. (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018), Vol. 107, pp. 1–11]. Our construction is obtained by combining tensor networks with expander codes [M. Sipser and D. Spielman, IEEE Trans. Inf. Theory 42, 1710 (1996)]. The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the “uncle Hamiltonian” of Fernández-González et al. [Commun. Math. Phys. 333, 299 (2015)]. Thus, we deviate from the quantum Calderbank-Shor-Steane (CSS) code Hamiltonians considered in most prior works.
Christophe Vuillot and Nikolas P. Breuckmann
Institute of Electrical and Electronics Engineers (IEEE)
We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a vast generalization of quantum color codes and Reed-Muller codes. A lot of the structure and properties of color codes carries over to pin codes. Pin codes have gauge operators, an unfolding procedure and their stabilizers form multi-orthogonal spaces. This last feature makes them interesting for devising magic-state distillation protocols. We study examples of these codes and their properties.
Nikolas P. Breuckmann and Vivien Londe
Institute of Electrical and Electronics Engineers (IEEE)
We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as <inline-formula> <tex-math notation="LaTeX">$n^\\epsilon $ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$\\epsilon > 0$ </tex-math></inline-formula> and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate <inline-formula> <tex-math notation="LaTeX">$k/n$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$13/72 = 0.180\\ldots $ </tex-math></inline-formula> and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.
Nikolas P. Breuckmann and Jens Niklas Eberhardt
American Physical Society (APS)
Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. We introduce the zoo of LDPC quantum codes and discuss their potential for making quantum computers robust against noise. In particular, we explain recent advances in the theory of LDPC quantum codes related to certain product constructions and discuss open problems in the field.
Nikolas P. Breuckmann and Jens N. Eberhardt
Institute of Electrical and Electronics Engineers (IEEE)
This work provides the first explicit and non-random family of <inline-formula> <tex-math notation="LaTeX">$[[N,K,D]]$ </tex-math></inline-formula> LDPC quantum codes which encode <inline-formula> <tex-math notation="LaTeX">$K \\in \\Theta \\left({N^{\\frac {4}{5}}}\\right)$ </tex-math></inline-formula> logical qubits with distance <inline-formula> <tex-math notation="LaTeX">$D \\in \\Omega \\left({N^{\\frac {3}{5}}}\\right)$ </tex-math></inline-formula>. The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called <italic>balanced product</italic>. Recently, Hastings–Haah–O’Donnell and Panteleev–Kalachev were the first to show that there exist families of LDPC quantum codes which break the <inline-formula> <tex-math notation="LaTeX">$\\mathrm {polylog}(N)\\sqrt {N}$ </tex-math></inline-formula> distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have <inline-formula> <tex-math notation="LaTeX">$K\\in \\Theta (N)$ </tex-math></inline-formula> and that we conjecture to have linear distance <inline-formula> <tex-math notation="LaTeX">$D\\in \\Theta (N)$ </tex-math></inline-formula>.
Oscar Higgott and Nikolas P. Breuckmann
American Physical Society (APS)
We introduce a technique that uses gauge fixing to significantly improve the quantum error correcting performance of subsystem codes. By changing the order in which check operators are measured, valuable additional information can be gained, and we introduce a new method for decoding which uses this information to improve performance. Applied to the subsystem toric code with three-qubit check operators, we increase the threshold under circuit-level depolarising noise from $0.67\\%$ to $0.81\\%$. The threshold increases further under a circuit-level noise model with small finite bias, up to $2.22\\%$ for infinite bias. Furthermore, we construct families of finite-rate subsystem LDPC codes with three-qubit check operators and optimal-depth parity-check measurement schedules. To the best of our knowledge, these finite-rate subsystem codes outperform all known codes at circuit-level depolarising error rates as high as $0.2\\%$, where they have a qubit overhead that is $4.3\\times$ lower than the most efficient version of the surface code and $5.1\\times$ lower than the subsystem toric code. Their threshold and pseudo-threshold exceeds $p=0.42\\%$ for circuit-level depolarising noise, increasing to $p=2.4\\%$ under infinite bias using gauge fixing.
Xingchuan Zhu, Jiaojiao Guo, Nikolas P Breuckmann, Huaiming Guo, and Shiping Feng
IOP Publishing
The effect of many-body interaction in curved space is studied based on the extended Bose–Hubbard model on hyperbolic lattices. Using the mean-field approximation and quantum Monte Carlo simulation, the phase diagram is explicitly mapped out, which contains the superfluid, supersolid and insulator phases at various fillings. Particularly, it is revealed that the sizes of the Mott lobes shrink and the supersolid is stabilized at smaller nearest-neighbor interaction as q in the Schläfli symbol increases. The underlying physical mechanism is attributed to the increase of the coordination number, and hence the kinetic energy and the nearest-neighbor interaction. The results suggest that the hyperbolic lattices may be a unique platform to study the effect of the coordination number on quantum phase transitions, which may be relevant to the experiments of ultracold atoms in optical lattices.
Jens N Eberhardt, Nikolas P Breuckmann, and Christiane S Eberhardt
Elsevier BV
J.N. Eberhardt, N.P. Breuckmann, and C.S. Eberhardt
Elsevier BV
Nikolas P. Breuckmann, Benedikt Placke, and Ananda Roy
American Physical Society (APS)
The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two- and three-dimensional hyperbolic spaces using Monte Carlo and high- and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the "asymptotic" limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three-dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.
K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal
Institute of Electrical and Electronics Engineers (IEEE)
We describe a computationally efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding a minimal surface given its boundary (curve) in any hypercubic lattice of dimension <inline-formula> <tex-math notation="LaTeX">$D>2$ </tex-math></inline-formula>. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of 4.35 ± 0.1%. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of 7.3 ± 0.1%. For the gate-based depolarizing error model, we find a threshold of 0.31 ± 0.01% which is below the threshold found for the two-dimensional toric code.
J. Conrad, C. Chamberland, N. P. Breuckmann, and B. M. Terhal
The Royal Society
We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for three-dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.
Nikolas P. Breuckmann and Xiaotong Ni
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.
Nikolas P Breuckmann, Christophe Vuillot, Earl Campbell, Anirudh Krishna, and Barbara M Terhal
IOP Publishing
We show how a hyperbolic surface code could be used for overhead-efficient quantum storage. We give numerical evidence for a noise threshold of for the -hyperbolic surface code in a phenomenological noise model (as compared with for the toric code). In this code family, parity checks are of weight 4 and 5, while each qubit participates in four different parity checks. We introduce a family of semi-hyperbolic codes that interpolate between the toric code and the -hyperbolic surface code in terms of encoding rate and threshold. We show how these hyperbolic codes outperform the toric code in terms of qubit overhead for a target logical error probability. We show how Dehn twists and lattice code surgery can be used to read and write individual qubits to this quantum storage medium.
Nikolas P. Breuckmann and Barbara M. Terhal
Institute of Electrical and Electronics Engineers (IEEE)
We show how to obtain concrete constructions of homological quantum codes based on tilings of 2-D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in 2-D quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction.
Nikolas P Breuckmann and Barbara M Terhal
IOP Publishing
The circuit-to-Hamiltonian construction translates dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman–Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each interacting qubit. This makes it possible to capture the spatio-temporal structure of the original quantum circuit into features of the circuit Hamiltonian. The construction is inspired by the original two-dimensional interacting fermion model in Mizel et al (2001 Phys. Rev. A 63 040302). We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lowerbounded so that the applications of this construction for quantum Merlin–Arthur (and partially for quantum adiabatic computation) go through. For one-dimensional quantum circuits, the dynamics generated by the circuit Hamiltonian corresponds to the diffusion of a string around the torus.
RECENT SCHOLAR PUBLICATIONS
MOST CITED SCHOLAR PUBLICATIONS
GRANT DETAILS
Co-applicant Making noisy quantum processors practical'' iUK-NSERC Canada-UK Quantum Technologies [GBP 300,000 in funds] (2021)
UCLQ Post-Doctoral Research Fellowship in Quantum Technologies [GBP 235,640 in funds] (awarded 2017, deferred for one year)
HPC project of 1.3 million core-hours at the RWTH Compute Cluster to simulate the performance of quantum fault-tolerance schemes (2016)
Travel grant to visit the conference QStart at Hebrew University Jerusalem [EUR 1,000 in funds] (2013)
RESEARCH OUTPUTS (PATENTS, SOFTWARE, PUBLICATIONS, PRODUCTS)
U.S. Patent Application No. 17/444 943, August 2021
INDUSTRY EXPERIENCE
PsiQuantum