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arXiv:2501.16028

Remarks on controlled measurement and quantum algorithm for calculating Hermitian conjugate

Edward B. Fel’dman, Alexander Ivanovich Zenchuk, Wentao Qi, Junde Wu

Published Jan 27, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2501.16028

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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.
In this paper, we solve three important problems that are revealed, in particular, to matrix-manipulation algorithms. The principal novelty is introducing the concept of quantumly controlled measurement that removes the post-selection problem by solving the problem of small access probability to the desired state of ancilla and possesses several remarkable properties. We also introduce separate encoding of the real and imaginary parts of a complex matrix that allows to include the Hermitian conjugation into the list of matrix manipulations. Finally, we weaken the constraints on the { modulus} of matrix elements unavoidably imposed by the normalization condition for a pure quantum state. The quantumly controlled measurement together with both other extensions are implemented into the matrix multiplication algorithm. The appropriate circuits are presented.
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Abstract We propose the variational quantum singular value decomposition based on encoding the elements of the considered N × N matrix into the state of a quantum system of appropriate dimension. This method doesn’t use the expansion of this matrix in terms of the unitary matrices. Controlled measurement is involved to avoid small success probability in ancilla measurement. The objective function for maximization algorithm can be obtained probabilistically via measurement of the states of two one-qubit subsystems. The circuit requires O (log N ) qubits for realization of this algorithm whose depths is proportional to log N / ε , where ε is the precision required for calculation of singular values.
We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary N×NN\times NN×N complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new values of optimization parameters for the next iteration of the loss function in the gradient optimization method. The depth and size of the circuit for this algorithm are, respectively, O(N2log⁡N)O(N^2 \log N)O(N2logN) and O(log⁡N)O(\log N)O(logN).
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)O(1)O(1) and that of other protocols logarithmically increases with the dimensionality of the considered matrices.
Abstract We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an ( N – 1) × ( N – 1) matrix (depth is O ( N 2 log N )) which is a simple modification of the algorithm for calculating the determinant of an N × N matrix (depth is O ( N log 2 N )). The basic idea is to encode each row of the matrix into a pure state of some quantum system. In addition, we use the representation of the elements of the inverse matrix in terms of algebraic complements. This algorithm together with that for matrix multiplication proposed earlier yields the algorithm for solving systems of linear algebraic equations (depth is O ( N log 2 N )). Measurement of the ancilla state with output 1 (probability is ~ 2 − O ( N log N ) ) removes the garbage acquired during calculation. Appropriate circuits for all three algorithms are presented and have the same estimation O ( N log N ) for the space (number of qubits in the circuit).