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arXiv:2311.11329 · Quantum Information and Computation

Matrix manipulations via unitary transformations and ancilla-state measurements

Alexander Ivanovich Zenchuk, Wentao Qi, Asutosh Kumar, Junde Wu

Published Nov 1, 2024• 5 citations
arXivHTMLPDF

Abstract

We propose protocols for calculating inner product, matrix addition and matrix multiplication based on multiqubit Toffoli-type and the simplest one-qubit operations and employ ancilla-measurements to remove all garbage of calculations. The depth (runtime) of addition protocol is $O(1)$ and that of other protocols logarithmically increases with the dimensionality of the considered matrices.

DOI
10.26421/qic24.13-14-2

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We present two new aspects for the recently proposed algorithms for matrix manipulating based on the special encoding the matrix elements into the superposition state of a quantum system. First aspect is the controlled measurement which allows to avoid the problem of small access probability to the required ancilla state at the final step of algorithms needed to remove the garbage of the states. Application of controlled measurement to the earlier developed algorithm is demonstrated. The second aspect is the algorithm for calculating the Hermitian conjugate of an arbitrary matrix, which supplements the algorithms proposed earlier. The appropriate circuits are presented.
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The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. In quantum computation, matrices ‘operate’ or act on vectors to move a system from one quantum state to another as it evolves with time. The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. Matrices can also be used to perform transformations of points in two- and three-dimensional space through rotations and reflections. Self-adjoint matrices play an important role in quantum computation because, as we shall see, they have real eigenvalues which correspond to the measured values of quantum observables. The procedure for calculating the inverse of a general n×n matrix necessitates the introduction of some new terminology. The rows of a unitary matrix are pairwise orthogonal so that the inner product of two distinct rows will be zero.
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics—with the result encoded in the off-diagonal blocks of a Hamiltonian—using Hamiltonian evolutions of input operators. We show how to maintain this Hamiltonian block encoding , so that matrix operations can be composed one after another, and the entire quantum computation takes ≤ 2 ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with 1 ancilla qubit.