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Physical Review A

Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

Jacob Chevalier Drori, Yordan S. Yordanov, T. Ferrus, M. J. Applegate, C. H. W. Barnes

Published Nov 24, 2021• 0 citations
Publisher

Abstract

We present a set of techniques, based on the repeated arbitrary application of cp, cnot, and swap${}^{\ensuremath{\alpha}}$ (power-of-swap) quantum gate operations to an $n$-qubit quantum computer that can be used in its verification. We find isomorphisms between the groups generated by these gate operations and known groups and use techniques from representation theory to determine their invariant subspaces. For the cp operation, we find an isomorphism to the direct product of $n(n\ensuremath{-}1)/2$ cyclic groups of order 2, and determine ${2}^{n}$ one-dimensional invariant subspaces corresponding to the computational-basis vectors. For the cnot operation, we find an isomorphism to $GL(n,2)$, and determine two one-dimensional invariant subspaces and one $({2}^{n}\ensuremath{-}2)$-dimensional invariant subspace. For the swap${}^{\ensuremath{\alpha}}$ operation we find a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. Using knowledge of these invariant subspaces, we propose a hardware verification scheme which tests the correct functioning of a quantum computer.

DOI
10.1103/physreva.104.052619

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We present a set of techniques, based on the repeated arbitrary application of cp, cnot, and swap\ensuremathα{}^{\ensuremath{\alpha}}\ensuremathα (power-of-swap) quantum gate operations to an nnn-qubit quantum computer that can be used in its verification. We find isomorphisms between the groups generated by these gate operations and known groups and use techniques from representation theory to determine their invariant subspaces. For the cp operation, we find an isomorphism to the direct product of n(n\ensuremath−1)/2n(n\ensuremath{-}1)/2n(n\ensuremath−1)/2 cyclic groups of order 2, and determine 2n{2}^{n}2n one-dimensional invariant subspaces corresponding to the computational-basis vectors. For the cnot operation, we find an isomorphism to GL(n,2)GL(n,2)GL(n,2), and determine two one-dimensional invariant subspaces and one (2n\ensuremath−2)({2}^{n}\ensuremath{-}2)(2n\ensuremath−2)-dimensional invariant subspace. For the swap\ensuremathα{}^{\ensuremath{\alpha}}\ensuremathα operation we find a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. Using knowledge of these invariant subspaces, we propose a hardware verification scheme which tests the correct functioning of a quantum computer.
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This project is a comprehensive investigation into the application of the exchange interaction, particularly with the realization of the SWAP^1/n quantum operator, in quantum information processing. We study the generation, characterization and application of entanglement in such systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical study of combinations of these gates is an interesting avenue of research that we have explored, though due to the exponential scaling of the complexity of the problem with the number of qubits in the system, numerical techniques, though good for few-qubit systems, are found to be inefficient for this research problem when we look at systems with higher number of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric group Sn, we employ group-theoretic methods to find the relevant invariant subspaces and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving symmetry of the vectors spanning the various invariant subspaces. We also devise new ways of characterizing entanglement and approach the separability problem by looking at permutation symmetries of subsystems of quantum states. This idea is found to form a bridge with the entanglement characterization tool of Peres-Horodecki’s Partial Positive Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented ‘distance’ measures of entanglement, besides devising a new entanglement witness in the ‘engle’. In terms of applications, we define five different formalisms for quantum computing: the circuit-based model, the encoded qubit model, the cluster-state model, functional quantum computation and the qudit-based model. Later in the thesis, we explore the idea of quantum computing based on decoherence-free subspaces. We also investigate ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum communication protocols and quantum memory.
All quantum gates with one and two qubits may be described by elements of Spin groups due to isomorphisms Spin(3)\isomSU(2) and Spin(6)\isomSU(4). However, the group of n-qubit gates SU(2^n) for n>2 has bigger dimension than Spin(3n). A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation SU(2^n). Analogously, the `Spin(3n)Spin(3n)Spin(3n) circuits' are introduced in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms. The matrix tensor product implementation of the Spin(3n) group together with relevant models by usual quantum circuits with 2n qubits are investigated in such a framework. A certain resemblance with well-known sets of non-universal quantum gates (e.g., matchgates, noninteracting-fermion quantum circuits) related with Spin(2n) may be found in presented approach. Finally, a possibility of the classical simulation of such circuits in polynomial time is discussed.
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Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
Pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a 4 × 4 matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of fourdimensional numbers, of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguish these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of fourdimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing.