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arXiv:2111.01211

Quantum Circuits in Additive Hilbert Space

Luca Mondada

Published Nov 1, 2021• 0 citations
arXivHTMLPDF

Abstract

Representations of quantum computations are almost always based on a tensor product $\otimes$-structure. This coincides with what we are able to execute in our experiments, as well as what we observe in Nature, but it makes certain familiar quantum primitives convoluted. Reversible classical circuits, diagonal operations or controlled unitaries all have very elegant and simple matrix representations that cannot be expressed succinctly as a circuit in a simple gate set, complicating quantum algorithm design and circuit optimization. We propose a new additive presentation of quantum computation to address this. We show how conventional circuits can be expressed in the additive space and how they can be recovered. In particular in our formalism we are able to synthesize high-level multi-controlled primitives from low-level circuit decompositions, making it an invaluable tool for circuit optimization. Our formulation also accepts a circuit-like diagrammatic representation and proposes a novel and simple interpretation of quantum computation.

DOI
10.48550/arxiv.2111.01211

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Representations of quantum computations are almost always based on a tensor product ⊗\otimes⊗-structure. This coincides with what we are able to execute in our experiments, as well as what we observe in Nature, but it makes certain familiar quantum primitives convoluted. Reversible classical circuits, diagonal operations or controlled unitaries all have very elegant and simple matrix representations that cannot be expressed succinctly as a circuit in a simple gate set, complicating quantum algorithm design and circuit optimization. We propose a new additive presentation of quantum computation to address this. We show how conventional circuits can be expressed in the additive space and how they can be recovered. In particular in our formalism we are able to synthesize high-level multi-controlled primitives from low-level circuit decompositions, making it an invaluable tool for circuit optimization. Our formulation also accepts a circuit-like diagrammatic representation and proposes a novel and simple interpretation of quantum computation.
Quantum computing is arguably one of the most revolutionary and disruptive technologies of this century. Due to the ever-increasing number of potential applications as well as the continuing rise in complexity, the development, simulation, optimization, and physical realization of quantum circuits is of utmost importance for designing novel algorithms. We show how matrix product states (MPSs) and matrix product operators (MPOs) can be used to express certain quantum states, quantum gates, and entire quantum circuits as low-rank tensors. This enables the analysis and simulation of complex quantum circuits on classical computers and to gain insight into the underlying structure of the system. We present different examples to demonstrate the advantages of MPO formulations and show that they are more efficient than conventional techniques if the bond dimensions of the wave function representation can be kept small throughout the simulation.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
Quantum circuit compilation is a critical step in realizing quantum computation, aiming to translate high-level quantum algorithms into a sequence of low-level gate operations executable on a quantum hardware. Traditional compilation methods often suffer from exponential scaling in the representation of the quantum state space, leading to prohibitive computational costs. This work introduces a novel approach to quantum circuit compilation utilizing tensor networks. We represent quantum circuits and their resulting states as higher-order tensor networks, enabling efficient manipulation and contraction of these representations. Our algorithm leverages tensor contraction and permutation techniques specifically tailored to the inherent structure of quantum circuits, minimizing the rank of the tensor networks and, consequently, reducing computational overhead. This approach offers the potential for significant speedups compared to conventional compilation techniques, addressing the core challenge of exponential scaling in quantum state representation. The method's effectiveness is demonstrated through theoretical analysis and a novel compilation algorithm.
Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing.Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically.However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2).Realizing any given single qubit gate using a given universal gate family is a complex task.Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation.Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm.The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately.The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm.The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm.Two specific directions of research have been studied.Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained.As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves.Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before.Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next.Extensive simulations show that the second version yields better results despite equivalent computation times.Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena.The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization.This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed.This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages.This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.
This chapter introduces Boolean quantum gates that allow us to map any Boolean circuit in such a way that the circuit becomes reversible. It describes quantum gates for one qubit, like for example Clifford gates that are the elements of the Clifford group and can be efficiently simulated with a classical computer. The chapter introduces the controlled-U gates and the unitary decomposition and formulate the process of transpilation. A Boolean quantum circuit represents a permutation in Hilbert space and can be represented by a unitary permutation matrix composed of the unitary matrices representing the quantum gates. Such a mapping does not alter the distribution of the amplitudes; the distribution remains unchanged during the execution of the quantum Boolean gates.
We conduct a systematic study of quantum circuits composed of multiple-control ZZZ-rotation (mczr) gates as primitives, since they are widely used components in quantum algorithms and also have attracted much experimental interest in recent years. We establish a circuit-polynomial correspondence to characterize the functionality of quantum circuits over the mczr gate set with continuous parameters. An analytic method for exactly synthesizing such quantum circuit to implement any given diagonal unitary matrix with an optimal gate count is proposed, which also enables the circuit depth optimal for specific cases with pairs of complementary gates. Furthermore, we present a gate-exchange strategy together with a flexible iterative algorithm for effectively optimizing the depth of any mczr circuit, which can also be applied to quantum circuits over any other commuting gate set. Besides the theoretical investigation, the practical performance of our circuit synthesis and optimization techniques is further evaluated numerically on two typical examples in quantum computing, including diagonal Hermitian operators and quantum approximate optimization algorithm circuits with tens of qubits, which can demonstrate a reduction in circuit depth by 33.40% and 15.55% on average over relevant prior works, respectively. Therefore, our methods and results provide a pathway for implementing quantum circuits and algorithms on recently developed devices.