arXiv subjects

Alex Bocharov

Publications and source records attributed to Alex Bocharov.

At least 19 recordsLinked to original sources

arXiv2025

Response to recent comments on Phys. Rev. B 107, 245423 (2023) and Subsection S4.3 of the Supp. Info. for Nature 638, 651-655 (2025)

The topological gap protocol (TGP) is a statistical test designed to identify a topological phase with high confidence and without human bias. It is used to determine a promising parameter regime for operating topological qubits. The protocol's key metric is the probability of incorrectly identifying a trivial region as topological, referred to as the false discovery rate (FDR). Two recent manuscripts [arXiv:2502.19560, arXiv:2503.08944] engage with the topological gap protocol and its use in Phys. Rev. B 107, 245423 (2023) and Subsection S4.3 of the Supplementary Information for Nature 638, 651-655 (2025), although they do not explicitly dispute the main results of either one. We demonstrate that the objections in arXiv:2502.19560 and arXiv:2503.08944 are unfounded, and we uphold the conclusions of Phys. Rev. B 107, 245423 (2023) and Nature 638, 651-655 (2025). Specifically, we show that no flaws have been identified in our estimate of the false discovery rate (FDR). We provide a point-by-point rebuttal of the comments in arXiv:2502.19560 and arXiv:2503.08944.

cond-mat.mes-hall↗

arXiv2025

Roadmap to fault tolerant quantum computation using topological qubit arrays

We describe a concrete device roadmap towards a fault-tolerant quantum computing architecture based on noise-resilient, topologically protected Majorana-based qubits. Our roadmap encompasses four generations of devices: a single-qubit device that enables a measurement-based qubit benchmarking protocol; a two-qubit device that uses measurement-based braiding to perform single-qubit Clifford operations; an eight-qubit device that can be used to show an improvement of a two-qubit operation when performed on logical qubits rather than directly on physical qubits; and a topological qubit array supporting lattice surgery demonstrations on two logical qubits. Devices that enable this path require a superconductor-semiconductor heterostructure that supports a topological phase, quantum dots and coupling between those quantum dots that can create the appropriate loops for interferometric measurements, and a microwave readout system that can perform fast, low-error single-shot measurements. We describe the key design components of these qubit devices, along with the associated protocols for demonstrations of single-qubit benchmarking, Clifford gate execution, quantum error detection, and quantum error correction, which differ greatly from those in more conventional qubits. Finally, we comment on implications and advantages of this architecture for utility-scale quantum computation.

quant-ph↗

arXiv2024

Interferometric Single-Shot Parity Measurement in an InAs-Al Hybrid Device

The fusion of non-Abelian anyons or topological defects is a fundamental operation in measurement-only topological quantum computation. In topological superconductors, this operation amounts to a determination of the shared fermion parity of Majorana zero modes. As a step towards this, we implement a single-shot interferometric measurement of fermion parity in indium arsenide-aluminum heterostructures with a gate-defined nanowire. The interferometer is formed by tunnel-coupling the proximitized nanowire to quantum dots. The nanowire causes a state-dependent shift of these quantum dots' quantum capacitance of up to 1 fF. Our quantum capacitance measurements show flux h/2e-periodic bimodality with a signal-to-noise ratio of 1 in 3.7 $\mu$s at optimal flux values. From the time traces of the quantum capacitance measurements, we extract a dwell time in the two associated states that is longer than 1 ms at in-plane magnetic fields of approximately 2 T. These results are consistent with a measurement of the fermion parity encoded in a pair of Majorana zero modes that are separated by approximately 3 $\mu$m and subjected to a low rate of poisoning by non-equilibrium quasiparticles. The large capacitance shift and long poisoning time enable a parity measurement error probability of 1%.

cond-mat.mes-hall↗

arXiv2022

Image Classification by Throwing Quantum Kitchen Sinks at Tensor Networks

Several variational quantum circuit approaches to machine learning have been proposed in recent years, with one promising class of variational algorithms involving tensor networks operating on states resulting from local feature maps. In contrast, a random feature approach known as quantum kitchen sinks provides comparable performance, but leverages non-local feature maps. Here we combine these two approaches by proposing a new circuit ansatz where a tree tensor network coherently processes the non-local feature maps of quantum kitchen sinks, and we run numerical experiments to empirically evaluate the performance of the new ansatz on image classification. From the perspective of classification performance, we find that simply combining quantum kitchen sinks with tensor networks yields no qualitative improvements. However, the addition of feature optimization greatly boosts performance, leading to state-of-the-art quantum circuits for image classification, requiring only shallow circuits and a small number of qubits -- both well within reach of near-term quantum devices.

quant-ph↗

arXiv2020

Predicting human-generated bitstreams using classical and quantum models

A school of thought contends that human decision making exhibits quantum-like logic. While it is not known whether the brain may indeed be driven by actual quantum mechanisms, some researchers suggest that the decision logic is phenomenologically non-classical. This paper develops and implements an empirical framework to explore this view. We emulate binary decision-making using low width, low depth, parameterized quantum circuits. Here, entanglement serves as a resource for pattern analysis in the context of a simple bit-prediction game. We evaluate a hybrid quantum-assisted machine learning strategy where quantum processing is used to detect correlations in the bitstreams while parameter updates and class inference are performed by classical post-processing of measurement results. Simulation results indicate that a family of two-qubit variational circuits is sufficient to achieve the same bit-prediction accuracy as the best traditional classical solution such as neural nets or logistic autoregression. Thus, short of establishing a provable "quantum advantage" in this simple scenario, we give evidence that the classical predictability analysis of a human-generated bitstream can be achieved by small quantum models.

quant-ph↗

arXiv2019

Optimizing Clifford gate generation for measurement-only topological quantum computation with Majorana zero modes

One of the main challenges for quantum computation is that while the number of gates required to perform a non-trivial quantum computation may be very large, decoherence and errors in realistic quantum architectures limit the number of physical gate operations that can be performed coherently. Therefore, an optimal mapping of the quantum algorithm into the physically available set of operations is of crucial importance. We examine this problem for a measurement-only topological quantum computer based on Majorana zero modes, where gates are performed through sequences of measurements. Such a scheme has been proposed as a practical, scalable approach to process quantum information in an array of topological qubits built using Majorana zero modes. Building on previous work that has shown that multi-qubit Clifford gates can be enacted in a topologically protected fashion in such qubit networks, we discuss methods to obtain the optimal measurement sequence for a given Clifford gate under the constraints imposed by the physical architecture, such as layout and the relative difficulty of implementing different types of measurements. Our methods also provide tools for comparative analysis of different architectures and strategies, given experimental characterizations of particular aspects of the systems under consideration. As a further non-trivial demonstration, we discuss an implementation of the surface code in Majorana-based topological qubits. We use the techniques developed here to obtain an optimized measurement sequence that implements the stabilizer measurements using only fermionic parity measurements on nearest-neighbor topological qubit islands.

quant-ph↗

arXiv2019

Quantum Language Processing

We present a representation for linguistic structure that we call a Fock-space representation, which allows us to embed problems in language processing into small quantum devices. We further develop a formalism for understanding both classical as well as quantum linguistic problems and phrase them both as a Harmony optimization problem that can be solved on a quantum computer which we show is related to classifying vectors using quantum Boltzmann machines. We further provide a new training method for learning quantum Harmony operators that describe a language. This also provides a new algorithm for training quantum Boltzmann machines that requires no approximations and works in the presence of hidden units. We additionally show that quantum language processing is BQP-complete, meaning that it is polynomially equivalent to the circuit model of quantum computing which implies that quantum language models are richer than classical models unless BPP=BQP. It also implies that, under certain circumstances, quantum Boltzmann machines are more expressive than classical Boltzmann machines. Finally, we examine the performance of our approach numerically for the case of classical harmonic grammars and find that the method is capable of rapidly parsing even non-trivial grammars. This suggests that the work may have value as a quantum inspired algorithm beyond its initial motivation as a new quantum algorithm.

quant-ph↗

arXiv2018

Circuit-centric quantum classifiers

The current generation of quantum computing technologies call for quantum algorithms that require a limited number of qubits and quantum gates, and which are robust against errors. A suitable design approach are variational circuits where the parameters of gates are learnt, an approach that is particularly fruitful for applications in machine learning. In this paper, we propose a low-depth variational quantum algorithm for supervised learning. The input feature vectors are encoded into the amplitudes of a quantum system, and a quantum circuit of parametrised single and two-qubit gates together with a single-qubit measurement is used to classify the inputs. This circuit architecture ensures that the number of learnable parameters is poly-logarithmic in the input dimension. We propose a quantum-classical training scheme where the analytical gradients of the model can be estimated by running several slightly adapted versions of the variational circuit. We show with simulations that the circuit-centric quantum classifier performs well on standard classical benchmark datasets while requiring dramatically fewer parameters than other methods. We also evaluate sensitivity of the classification to state preparation and parameter noise, introduce a quantum version of dropout regularisation and provide a graphical representation of quantum gates as highly symmetric linear layers of a neural network.

quant-ph↗

arXiv2017

Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures

We determine the cost of performing Shor's algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to emulate the standard binary version of the algorithm by encoding each qubit in a three-level system. We compare the two approaches and analyze the complexity of implementing Shor's period finding function in the two models. We also highlight the fact that the cost of achieving universality through magic states in MTQC architecture is asymptotically lower than in generic ternary case.

quant-ph↗

arXiv2016

Improved Quantum Ternary Arithmetics

Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only $O(\log\,n)$, while using with $O(n)$ ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate $C(X): |i,j\rangle \mapsto |i, j + δ_{i,2} \mod 3\rangle$ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 π\, i/9},1,e^{2 π\, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.

quant-ph↗

arXiv2016

A Note on Optimality of Quantum Circuits over Metaplectic Basis

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate $R=|0\rangle \langle 0| + |1\rangle \langle 1| - |2\rangle \langle 2|$. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any two-level Householder reflection to precision $\varepsilon$ by a metaplectic circuit of $R$-count at most $C \, \log_3(1/\varepsilon) + O(\log \log 1/\varepsilon)$ with $C=8$. A new result in this note takes the constant down to $C=5$ for non-exceptional target reflections under a certain credible number-theoretical conjecture. The new method increases the chances of obtaining a truly optimal circuit but may not guarantee the true optimality. Efficient approximations of an important ternary quantum gate proposed by Howard, Campbell and others is also discussed. Apart from this, the note is mostly didactical: we demonstrate how to leverage Lenstra's integer geometry algorithm from 1983 for circuit synthesis.

quant-ph↗

arXiv2016

Efficient Topological Compilation for Weakly-Integral Anyon Model

A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary $n$-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in $O(3^{2\,n} \, \log{1/\varepsilon})$ and entanglement cost that is exponential in $n$. Another flavor of the algorithm produces efficient circuits with upper complexity bound in $O(n\,3^{2\,n} \, \log{1/\varepsilon})$ and no additional entanglement cost.

quant-ph↗

arXiv2015

Efficient Approximation of Diagonal Unitaries over the Clifford+T Basis

We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+$T$ basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit. The equivalent $T$-count of the synthesized circuit is bounded by $k \, C_0 \log_2(1/\varepsilon) + E(n,k)$, where $k$ is the number of distinct phases in the diagonal $n$-qubit unitary, $\varepsilon$ is the desired precision, $C_0$ is a quality factor of the implementation method ($1<C_0<4$), and $E(n,k)$ is the total entanglement cost (in $T$ gates). We determine an optimal decision boundary in $(k,n,\varepsilon)$-space where our decomposition algorithm achieves lower entanglement cost than previous state-of-the-art techniques. Our method outperforms state-of-the-art techniques for a practical range of $\varepsilon$ values and diagonal operators and can reduce the number of $T$ gates exponentially in $n$ when $k << 2^n$.

quant-ph↗

arXiv2015

A Framework for Approximating Qubit Unitaries

We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+$π/12$ [arXiv:1409.3552], running on average in time polynomial in $O(\log(1/\varepsilon))$ (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer.

quant-ph↗

arXiv2014

Optimal Ancilla-free Pauli+V Circuits for Axial Rotations

Recently Neil Ross and Peter Selinger analyzed the problem of approximating z- rotations by means of single-qubit Clifford+T circuits. Their main contribution is a deterministic-search technique which allowed them to make approximating circuits shallower. We adapt the deterministic-search technique to the case of Pauli+V circuits and prove similar results. Because of the relative simplicity of the Pauli+V framework, we use much simpler geometric methods.

quant-ph↗

arXiv2014

Efficient synthesis of probabilistic quantum circuits with fallback

Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of $T$ gates of about $1/3$ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+$T$ gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford+$T$ gate set. The protocol, which we call Probabilistic Quantum Circuits with Fallback (PQF), implements a walk on a discrete Markov chain in which the target unitary is an absorbing state and in which transitions are induced by multi-qubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols terminate after a finite number of steps. Specifically, we apply our method to the Clifford+$T$, Clifford+$V$, and Clifford+$π/12$ gate sets to achieve decompositions with expected gate counts of $\log_b(1/\varepsilon)+O(\log(\log(1/\varepsilon)))$, where $b$ is a quantity related to the expansion property of the underlying universal gate set.

quant-ph↗

arXiv2014

Efficient synthesis of universal Repeat-Until-Success circuits

Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a $2.5$ times reduction in expected $T$-count over ancilla-free techniques for single-qubit unitary decomposition. However, the previously best known algorithm to synthesize RUS circuits requires exponential classical runtime. In this paper we present an algorithm to synthesize an RUS circuit to approximate any given single-qubit unitary within precision $\varepsilon$ in probabilistically polynomial classical runtime. Our synthesis approach uses the Clifford+$T$ basis, plus one ancilla qubit and measurement. We provide numerical evidence that our RUS circuits have an expected $T$-count on average $2.5$ times lower than the theoretical lower bound of $3 \log_2 (1/\varepsilon)$ for ancilla-free single-qubit circuit decomposition.

quant-ph↗

arXiv2013

Asymptotically Optimal Topological Quantum Compiling

In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations for non-abelian quasiparticles described by the Fibonacci anyon model. We develop a probabilistically polynomial algorithm that outputs a braid pattern to approximate a given single-qubit unitary to a desired precision. We also classify the single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern and present an efficient algorithm to produce their braid patterns. Our techniques produce braid patterns that meet the uniform asymptotic lower bound on the compiled circuit depth and thus are depth-optimal asymptotically. Our compiled circuits are significantly shorter than those output by prior state-of-the-art methods, resulting in improvements in depth by factors ranging from 20 to 1000 for precisions ranging between $10^{-10}$ and $10^{-30}$.

quant-ph↗