A paper from our Compositional Intelligence team which uses QNLP techniques in musical composition. This paper is part of the work accompanying the world’s first ever symposium on Quantum Computing and Music. There has been tremendous progress in Artificial Intelligence (AI) for music, in particular for musical composition and access to large databases for commercialisation through the Internet.
We have demonstrated a prototypical hybrid classical and quantum computational workflow for the quantification of protein-ligand interactions. The workflow combines the DMET embedding procedure with the VQE approach for finding molecular electronic ground states. This is the first application of real quantum computers to the calculation of protein-ligand binding energies.
In this paper, we represent all the elementary matrices of size 2m × 2m using algebraic ZX-calculus, then show their properties on inverses and transpose using rewriting rules of ZX-calculus. As a result, we are able to depict matrices of this size by string diagrams without resorting to a diagrammatic normal form.
We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists only if the graphs are fractionally isomorphic.
lambeq, the first high-level Python library for Quantum Natural Language Processing (QNLP), is presented in this paper. The open-source toolkit offers a detailed hierarchy of modules and classes implementing all stages of a pipeline for converting sentences to string diagrams, tensor networks, and quantum circuits ready to be used on a quantum computer.
We give a new type of qudit ZW-calculus which has generators and rewriting rules similar to that of the qubit ZW-calculus. Furthermore, we establish a translation between this qudit ZW-calculus and the qudit ZX-calculus which is universal. Therefore, this qudit ZW-calculus is also universal for pure qudit quantum computing.
We focus on the practical aspect of quantum computational calculations of solid-state crystalline materials based on a theory developed in our group by using real quantum hardware with noise mitigation techniques. We select two periodic systems with different levels of complexity for these calculations, the distorted hydrogen chain and the iron crystal in the BCC and FCC phases, and evaluate the ground state energies.
We show how, with deft management of the quadratic form expansion representation, we may simulate individual stabiliser operations in O(n2) time matching the overall complexity of other simulation techniques. Our techniques provide economies of scale in the time to simulate simultaneous measurements of all (or nearly all) qubits in the standard basis and allow single-qubit measurements with deterministic outcomes to be simulated in constant time.
This report describes the parsing problem for Combinatory Categorial Grammar (CCG), showing how a combination of Transformer-based neural models and a symbolic CCG grammar can lead to substantial gains over existing approaches. The report provides a minimal introduction to CCG and CCG parsing, with many pointers to the relevant literature. It then describes the CCG supertagging problem and some recent work from Tian et al. (2020) which applies Transformer-based models to supertagging with great effect.
We introduce “Q-marginals,” which are quantum states encoding some probability distribution in a manner suitable for use in Quantum Monte Carlo Integration (QMCI) and show that these can be prepared directly from a classical circuit sampling for the probability distribution of interest.
In this paper. we derive from simple and reasonable assumptions a Gaussian noise model for NISQ Quantum Amplitude Estimation (QAE). We provide results from QAE run on various IBM superconducting quantum computers and Honeywell’s H1 trapped-ion quantum computer to show that the proposed model is a good fit for real world experimental data.
Combinatorial optimisation models a vast range of industrial processes aiming at improving their efficiency. In this case study, we apply four variational quantum heuristics running on IBM’s superconducting quantum processors to the job shop scheduling problem. Our problem optimises a steel manufacturing process.
We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. We combine the variational approach with maximum likelihood amplitude estimation in VQAE.
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double D(G) symmetry, where G is a finite group.
To make combinatorial optimisation more efficient, we introduce the Filtering Variational Quantum Eigensolver, which utilises filtering operators to achieve faster and more reliable convergence to the optimal solution. We explore the use of causal cones to reduce the number of qubits required on a quantum computer. Our methods perform better than the original VQE algorithm and QAOA.
This work paves the way for a new theory of grammar that provides novel “grammatical truths.” We give a nogo-theorem for the fact that our wirings for words make no sense for preordered monoids, the form which grammatical calculi usually take. Instead, they require diagrams – or equivalently, (free) monoidal categories.
We have designed and developed a layer-two solution to secure the exchange of information between blockchain nodes over the internet and introduced a second signature in transactions using post-quantum keys. Our versatile solution can be applied to any blockchain network.
This paper proposes a method of quantum Monte-Carlo integration that retains the full quadratic quantum advantage without requiring any arithmetic or the quantum Fourier transform to be performed on the quantum computer. The heart of the proposed method is a Fourier series decomposition of the sum that approximates the expectation in Monte-Carlo integration, with each component then estimated individually using quantum amplitude estimation.
The DisCoCat model has proved to be a valuable tool for studying compositional aspects of language. However, the strong dependency of the model on a specific grammar formalism, gives rise to both theoretical and practical problems. We solve these problems by reformulating DisCoCat as a passage from Combinatory Categorial Grammar (CCG) to a category of semantics.
This paper provides a basis for a proof of completeness of qudit ZX-calcluli and their unified formalism qufinite ZX-calculus, which intuitively means quantum computing for qudits, or hybrid finite-dimensional quantum systems, can be done purely diagrammatically.
The ability to transfer coherent quantum information between systems is a fundamental component of quantum technologies and leads to coherent correlations within the global quantum process. Motivated by recent techniques in randomised benchmarking, we develop a range of results for efficient estimation of correlations within a bipartite quantum channel.
In a collaboration with CERN, we propose a dual-PQC GAN for generative modelling applications in High-Energy Physics. This development enables the generation of samples from an ensemble of typical images – something not possible with conventional qGAN architectures.
Diagrams are becoming a prominent tool in both machine learning (ML) and quantum computing. We adapt a key tool of ML, gradients to general diagrammatic theories. This will enable one to do (quantum) ML fully diagrammatically, substantially broadening the road towards general quantum advantage and quantum NLP in particular.
In this work, we propose quantum Born machines as variational distributions over discrete variables. Our techniques enable efficient variational inference with distributions beyond those that are efficiently representable on a classical computer.
In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum computers for datasets of size ≥ 100 sentences. We use representations for sentences that have a natural mapping to quantum circuits to implement and successfully train two NLP models that solve simple sentence classification tasks on quantum hardware.
This paper is a “spiritual child” of the 2005 lecture notes Kindergarten Quantum Mechanics, which showed how a simple, pictorial extension of Dirac notation allowed several quantum features to be easily expressed and derived, using language even a kindergartener can understand.
We prove that there is no quantum speed-up when using quantum Monte-Carlo integration to estimate the mean (and other moments) of analytically-defined log-concave probability distributions prepared as quantum states using the Grover-Rudolph method.
We provide conceptual and mathematical foundations for near-term quantum natural language processing (QNLP) and do so in quantum computer scientist-friendly terms. We opted for an expository presentation style and provide references for supporting empirical evidence and formal statements concerning mathematical generality.
We perform the first implementation of an NLP task on noisy intermediate-scale quantum (NISQ) hardware. Sentences are instantiated as parameterised quantum circuits. We encode word-meanings in quantum states and explicitly account for grammatical structure – which even in mainstream NLP is not commonplace – by faithfully hardwiring it as entangling operations.
We simulate the effects of different noise types in state preparation. We find that the inclusion of redundant parameterised gates makes the circuits more noise resilient. We also report a circuit-dependent noise threshold above which the optimisation can converge to states with largely different physical properties from the target state.
Parameterised quantum circuits are a promising technology for achieving a quantum advantage. An important application is the variational simulation of time evolution of quantum systems. To make the most of quantum hardware, variational algorithms need to be as hardware-efficient as possible. Here we present alternatives to the time-dependent variational principle that are hardware-efficient and do not require matrix inversion.
We present the first complete implementation of a randomness and privacy amplification protocol based on Bell tests. This allows the building of device-independent random number generators which output provably unbiased and private numbers. We then showcase our protocol on the quantum computers from the IBM-Q experience.
We have modified molecular VQE to work with Periodic Boundary Conditions (PBC), enabling us to simulate crystalline solids. We estimate the cost of these calculations and find them to be significantly more expensive than molecular VQE calculations, leading us to develop new methods to reduce both qubit count and circuit depth for NISQ deployment.
We demonstrate that the qubit-routing problem has a natural interpretation as a reinforcement learning problem. The results show state-of-the-art performance when qubit routing is treated as an abstracted problem and suggest that reinforcement learning may lead to further gains being made when addressing backend optimisation more generally.
We describe a compilation strategy for Variational Quantum Eigensolver (VQE) algorithms which use the Unitary Coupled Cluster (UCC) ansatz. This strategy reduces cx depth by 75.4% on average and by up to 89.9% compared to naive synthesis for a variety of molecules, qubit encodings and basis sets.
We propose “application-motivated” circuit classes for benchmarking: deep, shallow and square quantum circuits. Using systems made available by IBM Q, we examine their performance, showing that noise-aware compilation strategies may be beneficial and that device connectivity and noise levels play a crucial role in the performance of the system.
In this work, we describe a full-stack pipeline for natural language processing on near-term quantum computers, aka QNLP. The language-modelling framework we employ is that of compositional distributional semantics (DisCoCat). Within this model, the grammatical reduction of a sentence is interpreted as a diagram, encoding a specific interaction of words according to the grammar. This interaction, together with a specific choice of word embedding, realises the meaning of a sentence.
We introduce DisCoPy, an open source toolbox for computing with monoidal categories. The library provides an intuitive syntax for defining string diagrams and monoidal functors. Its modularity allows the efficient implementation of computational experiments in the various applications of category theory where diagrams have become a lingua franca.
We propose a new algorithm to synthesise quantum circuits for phase polynomials, which takes into account the qubit connectivity. This work focuses on the architectures of current NISQ devices. The resulting algorithm generates circuits with a smaller CNOT depth than those currently used in Staq and Tket, while improving the runtime with respect to the former.
We describe techniques to reduce the T-count based on the effective application of “spider nest identities,” easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits in run-times, which are typically less than the time required to make a fresh cup of coffee.
We present TKET, a quantum software development platform produced by Cambridge Quantum. The heart of TKET is a language-agnostic optimising compiler designed to generate code for a variety of NISQ devices, which has several features designed to minimise the influence of device error.
We introduce the asymmetric extension of the Quantum Symmetric Simple Exclusion Process which is a stochastic model of fermions on a lattice hopping with random amplitudes. We analytically show that the time-integrated current of fermions defines a height field which exhibits a quantum non-linear stochastic Kardar-Parisi-Zhang dynamics.
In this paper we explore different update mechanisms for DisCoCirc, in the case where meaning is encoded in density matrices, which come with several advantages as compared to vectors.
The variational quantum eigensolver (VQE) requires specification of symmetries that describe the system, e.g. spin state and number of electrons. This opens the possibility of using VQE to obtain excited states. In this paper, various unitary coupled cluster (UCC) ansätze applied to excited states are investigated, using quantum circuits to represent single reference and multi-reference wavefunctions.
We present a quantum algorithm to perform dynamical mean field theory (DMFT) calculations for condensed matter systems on currently available quantum computers and demonstrate it on two quantum hardware platforms.
Hybrid quantum-classical systems make it possible to utilise existing quantum computers to their fullest extent. Within this framework, parameterised quantum circuits can be regarded as machine learning models with remarkable expressive power. This Review presents the components of these models and discusses their application to a variety of data-driven tasks, such as supervised learning and generative modelling.
In this paper, we give an overview of the circuit optimisation methods used by TKET. We focus on a novel technique based around phase gadgets presented in ZX-calculus, which makes it easy to reason about them. Taking advantage of this, we present an efficient method to translate the phase gadgets back to ∧X gates and single qubit operations suitable for execution on a quantum computer with significant reductions in gate count and circuit depth.
We propose an efficient method for simultaneously optimising both the structure and parameter values of quantum circuits with only a small computational overhead. Shallow circuits that use structure optimisation perform significantly better than circuits that use parameter updates alone, making this method particularly suitable for noisy intermediate-scale quantum computers.
In this paper, we give a mathematical foundation, referred to as DisCoCirc, for how sentences interact in texts in order to produce the meaning of that text. First we revisit DisCoCat. While in DisCoCat all meanings are fixed as states (i.e. have no input), in DisCoCirc, word meanings correspond to a type, or system, and the states of this system can evolve. Sentences are gates within a circuit which update the variable meanings of those words.
In this technical note, we theoretically motivate and empirically validate an initialisation strategy which can resolve the barren plateau problem for practical applications. The technique involves randomly selecting some of the initial parameter values, then choosing the remaining values so that the circuit is a sequence of shallow blocks that each evaluates to the identity.
We introduce a new architecture-agnostic methodology for mapping abstract quantum circuits to realistic quantum computing devices with restricted qubit connectivity, as implemented by TKET. We present empirical results showing the effectiveness of this method in terms of reducing two-qubit gate depth and two-qubit gate count compared to other implementations.
Generative modelling is a flavour of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. This study represents the first successful training of a high-dimensional universal quantum circuit and highlights the promise and challenges associated with hybrid learning schemes.
In this work, we derive an adversarial algorithm for the problem of approximating an unknown quantum pure state. Although this could be done on universal quantum computers, the adversarial formulation enables us to execute the algorithm on near-term quantum computers.
Quantum circuits with hierarchical structure have been used to perform binary classification of classical data encoded in a quantum state. We demonstrate that more expressive circuits in the same family achieve better accuracy and can be used to classify highly entangled quantum states, for which there is no known efficient classical method.
Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterisation of quantum devices and to train shallow circuits for generative tasks.
In this work, we introduce the quantum-assisted Helmholtz machine: a hybrid quantum-classical framework with the potential of tackling high-dimensional real-world machine learning datasets on continuous variables. We use deep learning to extract a low-dimensional binary representation of data, suitable for processing on relatively small quantum computers. The quantum hardware and deep learning architecture work together to train an unsupervised generative model.
We analyse the readiness of quantum annealing machines for real world application problems. These are typically not random and have an underlying structure that is hard to capture in synthetic benchmarks, thus posing unexpected challenges for optimisation techniques. We present a comprehensive computational scaling analysis of fault diagnosis in digital circuits, considering architectures beyond D-wave quantum annealers.
In this paper, we give a universal completion of the ZX-calculus for the whole of pure qubit quantum mechanics. This proof is based on the completeness of another graphical language, the ZW-calculus, with direct translations between these two graphical systems.
We give a quantum algorithm for solving semidefinite programs (SDPs). The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular, it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.
We consider the computational task of determining whether or not a given table of possibilities constitutes a departure from possibilistic local realism. By considering the case in which one party has access to measurements with two outcomes and the other three, it is possible to see at exactly which point this task becomes computationally difficult.
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearised Boltzmann transport equation, also including the effect of non-equilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system.
We show that quantum backaction of weak measurement can be used for tailoring long-range correlations of ultracold fermions, realising quantum states with spatial modulations of the density and magnetisation, thus overcoming usual requirement for a strong interatomic interactions. We propose detection schemes for implementing antiferromagnetic states and density waves. We demonstrate that such long-range correlations cannot be realised with local addressing.
The quantum circuit model allows gates between any pair of qubits, yet physical instantiations allow only limited interactions. We address this problem by providing an interaction graph together with an efficient method for compiling quantum circuits so that gates are applied only locally.