Family Of Circuits Research Articles

Modular Synthesis of Efficient Quantum Uncomputation

A key challenge of quantum programming is uncomputation: the reversible deallocation of qubits. And while there has been much recent progress on automating uncomputation, state-of-the-art methods are insufficient for handling today’s expressive quantum programming languages. A core reason is that they operate on primitive quantum circuits, while quantum programs express computations beyond circuits, for instance, they can capture families of circuits defined recursively in terms of uncomputation and adjoints. In this paper, we introduce the first modular automatic approach to synthesize correct and efficient uncomputation for expressive quantum programs. Our method is based on two core technical contributions: (i) an intermediate representation (IR) that can capture expressive quantum programs and comes with support for uncomputation, and (ii) modular algorithms over that IR for synthesizing uncomputation and adjoints. We have built a complete end-to-end implementation of our method, including an implementation of the IR and the synthesis algorithms, as well as a translation from an expressive fragment of the Silq programming language to our IR and circuit generation from the IR. Our experimental evaluation demonstrates that we can handle programs beyond the capabilities of existing uncomputation approaches, while being competitive on the benchmarks they can handle. More broadly, we show that it is possible to benefit from the greater expressivity and safety offered by high-level quantum languages without sacrificing efficiency.

Uncloneable Quantum Advice

The famous no-cloning principle has been shown recently to enable a number of uncloneable cryptographic primitives, including the copy-protection of certain functionalities. Here we address for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation, but that is natively unkeyed: quantum advice. Remarkably, this is an application of the no-cloning principle in a context where the quantum states of interest are not chosen by a random process. We establish unconditional constructions for promise problems admitting uncloneable quantum advice and, assuming the feasibility of quantum copy-protecting certain functions, for languages with uncloneable advice. Along the way, we note that state complexity classes, introduced by Rosenthal and Yuen (ITCS 2022) — which concern the computational difficulty of synthesizing sequences of quantum states — can be naturally generalized to obtain state cloning complexity classes. We make initial observations on these classes, notably obtaining a result analogous to the existence of undecidable problems. Our proof technique defines and constructs ingenerable sequences of finite bit strings, essentially meaning that they cannot be generated by any uniform circuit family with non-negligible probability. We then prove a generic result showing that the difficulty of accomplishing a computational task on uniformly random inputs implies its difficulty on any fixed, ingenerable sequence. We use this result to derandomize quantum cryptographic games that relate to cloning, and then incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable advice. Applying this two-step process to a monogamy-of-entanglement game yields a promise problem with uncloneable advice, and applying it to the quantum copy-protection of pseudorandom functions with super-logarithmic output lengths yields a language with uncloneable advice.

Quantum Information Spreading in Generalized Dual-Unitary Circuits.

We study the spreading of quantum information in a recently introduced family of brickwork quantum circuits that generalizes the dual-unitary class. These circuits are unitary in time, while their spatial dynamics is unitary only in a restricted subspace. First, we show that local operators spread at the speed of light as in dual-unitary circuits, i.e., the butterfly velocity takes the maximal value allowed by the geometry of the circuit. Then, we prove that the entanglement spreading can still be characterized exactly for a family of compatible initial states (in fact, for an extension of the compatible family of dual-unitary circuits) and that the asymptotic entanglement slope is again independent on the Rényi index. Remarkably, however, we find that the entanglement velocity is generically smaller than 1. We use these properties to find a closed-form expression for the entanglement-membrane line tension.

The Goldilocks principle of learning unitaries by interlacing fixed operators with programmable phase shifters on a photonic chip

Programmable photonic integrated circuits represent an emerging technology that amalgamates photonics and electronics, paving the way for light-based information processing at high speeds and low power consumption. Programmable photonics provides a flexible platform that can be reconfigured to perform multiple tasks, thereby holding great promise for revolutionizing future optical networks and quantum computing systems. Over the past decade, there has been constant progress in developing several different architectures for realizing programmable photonic circuits that allow for realizing arbitrary discrete unitary operations with light. Here, we systematically investigate a general family of photonic circuits for realizing arbitrary unitaries based on a simple architecture that interlaces a fixed intervening layer with programmable phase shifter layers. We introduce a criterion for the intervening operator that guarantees the universality of this architecture for representing arbitrary N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N \ imes N$$\\end{document} unitary operators with N+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N+1$$\\end{document} phase layers. We explore this criterion for different photonic components, including photonic waveguide lattices and meshes of directional couplers, which allows the identification of several families of photonic components that can serve as the intervening layers in the interlacing architecture. Our findings pave the way for efficiently designing and realizing novel families of programmable photonic integrated circuits for multipurpose analog information processing.

Absence of localization in weakly interacting Floquet circuits

We present a family of Floquet circuits that can interpolate between noninteracting qubits, free propagation, generic interacting, and dual-unitary dynamics. We identify the operator entanglement entropy of the two-qubit gate as a good quantitative measure of the interaction strength. We test the persistence of localization in the vicinity of the noninteracting point by probing spectral statistics, decay of autocorrelators, and measuring entanglement growth. The finite-size analysis suggests that the many-body localized regime does not persist in the thermodynamic limit. Instead, our results are compatible with an integrability-breaking phenomenon. Published by the American Physical Society 2024

Dual symplectic classical circuits: An exactly solvable model of many-body chaos

We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for a choice of observables.

A Case for Synthesis of Recursive Quantum Unitary Programs

Quantum programs are notoriously difficult to code and verify due to unintuitive quantum knowledge associated with quantum programming. Automated tools relieving the tedium and errors associated with low-level quantum details would hence be highly desirable. In this paper, we initiate the study of program synthesis for quantum unitary programs that recursively define a family of unitary circuits for different input sizes, which are widely used in existing quantum programming languages. Specifically, we present QSynth, the first quantum program synthesis framework, including a new inductive quantum programming language, its specification, a sound logic for reasoning, and an encoding of the reasoning procedure into SMT instances. By leveraging existing SMT solvers, QSynth successfully synthesizes ten quantum unitary programs including quantum adder circuits, quantum eigenvalue inversion circuits and Quantum Fourier Transformation, which can be readily transpiled to executable programs on major quantum platforms, e.g., Q#, IBM Qiskit, and AWS Braket.

Quantum Circuit Design for Integer Multiplication Based on Schönhage–Strassen Algorithm

Quantum arithmetic circuits has attracted extensive attention recently since it plays fundamental roles in many applications of quantum computing. Specifically, quantum circuits for integer multiplication are of great significance to various quantum algorithms including Shor’s integer factorization and discrete logarithm algorithm. In this paper, we design a family of quantum circuits for integer multiplication based on the famous classical integer multiplication algorithm, Schönhage-Strassen algorithm. We have made slight modifications to the algorithm to simplify its quantum circuit implementation. As a result, the quantum circuit we designed has gate depth O(log2n). To the best of our knowledge, this is the first poly-logarithmic depth quantum circuit for integer multiplication which keeps the circuit size and the number of ancillary qubits sub-quadratic. Our design has size O(nlognloglogn) counted by elementary quantum gates which is the same as the time complexity of Schönhage-Strassen algorithm, and it consumes O(nlognloglogn) clean ancillary qubits. In addition, we also utilize a weaker version of Schönhage-Strassen algorithm to give a family of circuits which has depth at the same order O(log2n) but with significantly smaller constants, while still keeping the size and number of ancillary qubits sub-quadratic.

An exponentially-growing family of universal quantum circuits

Quantum machine learning has become an area of growing interest but has certain theoretical and hardware-specific limitations. Notably, the problem of vanishing gradients, or barren plateaus, renders the training impossible for circuits with high qubit counts, imposing a limit on the number of qubits that data scientists can use for solving problems. Independently, angle-embedded supervised quantum neural networks were shown to produce truncated Fourier series with a degree directly dependent on two factors: the depth of the encoding and the number of parallel qubits the encoding applied to. The degree of the Fourier series limits the model expressivity. This work introduces two new architectures whose Fourier degrees grow exponentially: the sequential and parallel exponential quantum machine learning architectures. This is done by efficiently using the available Hilbert space when encoding, increasing the expressivity of the quantum encoding. Therefore, the exponential growth allows staying at the low-qubit limit to create highly expressive circuits avoiding barren plateaus. Practically, parallel exponential architecture was shown to outperform the existing linear architectures by reducing their final mean square error value by up to 44.7% in a one-dimensional test problem. Furthermore, the feasibility of this technique was also shown on a trapped ion quantum processing unit.

BioSFQ Circuit Family for Neuromorphic Computing: Bridging Digital and Analog Domains of Superconductor Technologies

Superconductor single flux quantum (SFQ) technology is attractive for neuromorphic computing due to low energy dissipation and high, potentially up to 100 GHz, clock rates. We have recently suggested a new family of bioSFQ circuits (V.K. Semenov <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> ., <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE TAS</i> , vol. 32, no. 4, 1400105, 2022) where information is stored as a value of current in a superconducting loop and transferred as a rate of SFQ pulses propagating between the loops. This approach, in the simplest case dealing with positive numbers, requires single-line transfer channels. In the more general case of bipolar numbers, it requires dual-rail transfer channels. To address this need, we have developed a new comparator with a dual-rail output. This comparator is an essential part of a bipolar multiplier performing an <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>X Y</i></b> operation on two analog currents <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>X</i></b> and <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>Y</i></b> . The multiplier has been designed, fabricated, and tested. We also present bioSFQ circuits for implementing an analog bipolar divide operation <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>Y</i>/<i>X</i></b> and a square root operation √ <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>X</i></b> . We discuss strategic advantages of the suggested bioSFQ approach, e.g., an inherently asynchronous character of bioSFQ cells which do not require explicit clock signals. As a result, bioSFQ circuits are free of racing errors and tolerant to occasional collision of propagating SFQ pulses. This tolerance is due to stochastic nature of data signals generated by comparators operating within their gray zone. The circuits were fabricated in the eight-niobium-layer fabrication process SFQ5ee developed for superconductor electronics at MIT Lincoln Laboratory.

Exploring Quantum Average-Case Distances: Proofs, Properties, and Examples

In this work, we present an in-depth study of average-case quantum distances introduced in [1]. The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average-case</i> statistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an δ-approximate 4-design, with δ = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–8</sup> ), then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that for each dimension <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> this ratio is at most <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.

Post-selection-free preparation of high-quality physical qubits

Rapidly improving gate fidelities for coherent operations mean that errors in state preparation and measurement (SPAM) may become a dominant source of error for fault-tolerant operation of quantum computers. This is particularly acute in superconducting systems, where tradeoffs in measurement fidelity and qubit lifetimes have limited overall performance. Fortunately, the essentially classical nature of preparation and measurement enables a wide variety of techniques for improving quality using auxiliary qubits combined with classical control and post-selection. In practice, however, post-selection greatly complicates the scheduling of processes such as syndrome extraction. Here we present a family of quantum circuits that prepare high-quality |0&amp;#x27E9; states without post-selection, instead using CNOT and Toffoli gates to non-linearly permute the computational basis. We find meaningful performance enhancements when two-qubit gate fidelities errors go below 0.2%, and even better performance when native Toffoli gates are available.

Efficient deterministic MapReduce algorithms for parallelizable problems

The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.'s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/log⁡n) rounds, where n is the input size and T(n) is the depth of the witnessing circuit family. Thus, we are able to closely relate the standard, uniform NC hierarchy modeling parallel computations to the deterministic MapReduce hierarchy DMRC by proving that NCi+1⊆DMRCi for all i∈N. Besides the theoretical significance, this result has important applied aspects as well. In particular, we show for all problems in NC1—many practically relevant ones, such as integer multiplication and division, the parity function, and recognizing balanced strings of parentheses being among these—how to solve them in a constant number of deterministic MapReduce rounds.

Unified multivariate trace estimation and quantum error mitigation

Calculating the trace of the product of $m\phantom{\rule{4pt}{0ex}}n$-qubit density matrices (multivariate trace) is a crucial subroutine in quantum error mitigation and information measure estimation. We propose a unified multivariate trace estimation (UMT) which conceptually unifies the previous qubit-optimal and depth-optimal approaches with tunable quantum circuit depth and number of qubits. The constructed circuits have $\ensuremath{\lceil}(m\ensuremath{-}1)/s\ensuremath{\rceil}$ or $n\ensuremath{\lceil}(m\ensuremath{-}1)/s\ensuremath{\rceil}$ depth corresponding to $(s+m)n$ or $s+mn$ qubits for $s\ensuremath{\in}{1,\ensuremath{\cdots},\ensuremath{\lfloor}m/2\ensuremath{\rfloor}}$, respectively. Such flexible circuit structures enable people to choose suitable circuits for different hardware devices. We apply UMT to virtual distillation to achieve exponential error suppression and design a family of concrete circuits to calculate the trace of the product of eight and nine $n$-qubit density matrices. A numerical example shows that the additional circuits still mitigate the noise expectation value under the global depolarizing channel.

Possibilistic simulation of quantum circuits by classical circuits

In a breakthrough work, Bravyi, Gosset, and K\"onig (BGK) [Science 362, 308 (2018)] unconditionally proved that constant-depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant-depth quantum circuits takes classical circuits at least $\mathrm{\ensuremath{\Omega}}(logn)$ depth to ``simulate,'' in a certain sense. In our paper, we formalize their sense of simulation, which we call ``possibilistic simulation'' or ``$p$-simulation,'' and construct explicit classical circuits that can $p$-simulate any depth-$d$ quantum circuit with Clifford and $t\phantom{\rule{4pt}{0ex}}T$-gates in depth $O(d+t)$. Our classical circuits use ${\mathrm{NOT},\mathrm{AND},\mathrm{OR}}$ gates of fan-in $\ensuremath{\le}2$.

Logical computation with self-assembling electric circuits.

Inspired by self-assembled biological growth, the Circuit Tile Assembly Model (cTAM) was developed to provide insights into signal propagation, information processing, and computation in bioelectric networks. The cTAM is an abstract model that produces a family of circuits of different sizes that is amenable to exact analysis. Here, the cTAM is extended to the Boolean Circuit Tile Assembly Model (bcTAM) that implements a computationally complete set of Boolean gates through self-assembled and self-controlled growth. The proposed model approximates axonal growth in neural networks and thus, investigates the computational capability of dynamic biological networks, for example, in growing networks of axons. Thus, the bcTAM models the effect of electrical activity on growth and shows how that growth might implement Boolean computations. In this sense, given a set of input voltages, the bcTAM is a system that is able to monitor and make decisions about its own growth.

Linear Dependent Type Theory for Quantum Programming Languages

Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M by Rios and Selinger (2017) constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.

Formal Language Recognition by Hard Attention Transformers: Perspectives from Circuit Complexity

Abstract This paper analyzes three formal models of Transformer encoders that differ in the form of their self-attention mechanism: unique hard attention (UHAT); generalized unique hard attention (GUHAT), which generalizes UHAT; and averaging hard attention (AHAT). We show that UHAT and GUHAT Transformers, viewed as string acceptors, can only recognize formal languages in the complexity class AC0, the class of languages recognizable by families of Boolean circuits of constant depth and polynomial size. This upper bound subsumes Hahn’s (2020) results that GUHAT cannot recognize the DYCK languages or the PARITY language, since those languages are outside AC0 (Furst et al., 1984). In contrast, the non-AC0 languages MAJORITY and DYCK-1 are recognizable by AHAT networks, implying that AHAT can recognize languages that UHAT and GUHAT cannot.

Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.

Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms

Variational quantum algorithms are considered to be appealing applications of near-term quantum computers. However, it has been unclear whether they can outperform classical algorithms or not. To reveal their limitations, we must seek a technique to benchmark them on large-scale problems. Here we propose a perturbative approach for efficient benchmarking of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which is enabled by exploiting the classical simulatability of Clifford circuits. Our method can be applied to a wide family of parameterized quantum circuits consisting of Clifford gates and single-qubit rotation gates. The approximate optimal parameter obtained by the method can also serve as an initial guess for further optimizations on a quantum device. As the first application of the method, we perform a benchmark of so-called hardware-efficient-type ansatzes when they are applied to the variational quantum eigensolver (VQE) of one-dimensional hydrogen chains up to ${\mathrm{H}}_{24}$, which corresponds to a 48-qubit system using a standard workstation. This is the largest scale benchmark of the VQE to the best of our knowledge and reveals the limitation of hardware-efficient-type ansatzes.