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arXiv:2212.12934 · Physical Review A

Near-optimal quantum circuit construction via Cartan decomposition

Maximilian Balthasar Mansky, Santiago Londoño Castillo, Victor Ramos Puigvert, Claudia Linnhoff–Popien

Published Nov 15, 2023• 9 citations
arXivHTMLPDF

Abstract

We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of $\frac{21}{16}{4}^{n}$ for $n$ qubits.

DOI
10.1103/physreva.108.052607

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A critical step in developing circuits for quantum simulation is to synthesize a desired unitary operator using the circuit building blocks. Studying unitaries and their generators from the Lie algebraic perspective has given rise to several algorithms for synthesis based on a Cartan decomposition of the dynamical Lie algebra. For unitaries of the form e−itHe^{-itH}e−itH, such as time-independent Hamiltonian simulation, the resulting circuits have depth that does not depend on simulation time ttt. However, finding such circuits has a large classical overhead in the cost function evaluation and the high dimensional optimization problem. In this work, by further partitioning the dynamical Lie algebra, we break down the optimization problem into smaller independent subproblems. Moreover, the resulting algebraic structure allows us to easily shift the evaluation of the cost function to the quantum computer, further cutting the classical overhead of the algorithm. As an application of the new hybrid algorithm, we synthesize the time evolution unitary for the 4-site transverse field Ising model on several IBM devices and Quantinuum's H1-1 quantum computer.
Abstract Quantum circuits of a general quantum gate acting on multiple ‐level quantum systems play a prominent role in multi‐valued quantum computation. A recursive Cartan decomposition of semi‐simple unitary Lie group (arbitrary ‐qutrit gate) is first proposed with a rigorous proof, which completely decomposes an ‐qutrit gate into local and non‐local operations. On this basis, an explicit quantum circuit is designed for implementing arbitrary two‐qutrit gates, and the cost of the construction is 21 generalized controlled (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Furthermore, the program is extended to the ‐qutrit system, and the quantum circuit of generic ‐qutrit gates contained GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far and its strength becomes more remarkable as increases, for example, when , the program saves 7146 GCXs compared to the previous best result. In addition, concrete recursive decomposition expressions is given for each non‐local operation instead of only quantum circuit diagrams.
Quantum circuits of a general quantum gate acting on multiple ddd-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group U(3n)U(3^n)U(3n) (arbitrary nnn-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the nnn-qutrit system, and the quantum circuit of generic nnn-qutrit gates contained 4196⋅32n−4⋅3n−1−(n22+n4−2932)\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})9641​⋅32n−4⋅3n−1−(2n2​+4n​−3229​) GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.
Cartan-Khaneja-Glaser decomposition of SU(2n)SU(2^n)SU(2n) via involutive automorphisms
We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in SU(2n)\mathrm{SU}\left(2^n\right)SU(2n), a critical task for efficient quantum circuit design. Building upon the approach introduced by Sá Earp and Pachos (2005), we overcome key limitations of their method, such as reliance on ill-defined matrix logarithms and the convergence issues of truncated Baker-Campbell-Hausdorff (BCH) series. Our reformulation leverages the algebraic structure of involutive automorphisms and symmetric Lie algebra decompositions to yield a stable and recursive factorization process. We provide a full Python implementation of the algorithm, available in an open-source repository, and validate its performance on matrices in SU(8)\mathrm{SU}(8)SU(8) and SU(16)\mathrm{SU}(16)SU(16) using random unitary benchmarks. The algorithm produces decompositions that are directly suited to practical quantum hardware, with factors that can be implemented near-optimally using standard gate sets.
We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with 2n2^n2n degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of 4n−2n4^n-2^n4n−2n rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.
This paper proposes an optimized quantum block-ZXZZXZZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.