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arXiv:2504.00015

Quantumly controlled measurement, Hermitian conjugation and normalization in matrix-manipulation algorithms

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

Published Mar 27, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2504.00015

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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.
Abstract We propose an algorithm to create an arbitrary n -qubit pure quantum superposition state with precision of m -decimals (binary representation) for each probability amplitude. The algorithm uses one-qubit rotations, Hadamard transformations and, C-NOT operations with multi-qubit controls. The depth of the circuit is O (2 n n ), its space is O ( n ). We emphasize that the parameters of the utilized unitary transformations are predicted in advance by the required precision and therefore there is no classical calculation supplementing this quantum algorithm. Finalizing the state-creation we perform the measurement of the ancilla state with a certain desired output with the purpose of removing all the garbage from the created superposition state. If n and m are independent and n ≫ m , the probability of access to the desired ancilla-state (success probability) is ~ 2 −n , which requires O (2 n ) runs of the algorithm and leads to the overall depth O (4 n n ). However, the situation can be significantly improved by replacing the usual measurement with the controlled measurement of the ancilla state that, first of all, removes the garbage part of the superposition state and, second (and most important), allows to avoid both the problem of small success probability to the desired ancilla state and multiple runs of the algorithm. As a consequence, the depth of the algorithm is O (2 n n ) in this case. If m serves to provide the required fidelity of state approximation, then the above parameters increase. This algorithm can be a subroutine generating the required input state in various algorithms, in particular, in matrix-manipulation algorithms developed earlier.
Algorithms based on non-unitary evolution have attracted much interest for ground state preparation on quantum computers. One recently proposed method makes use of ancilla qubits and controlled unitary operators to implement weak measurements related to imaginary-time evolution. In this work we specialize and extend this approach to the setting of combinatorial optimization. We first generalize the algorithm from exact to approximate optimization, taking advantage of several properties unique to classical problems. In particular we show how to select parameters such that the success probability of each measurement step is bounded away from 1/21/21/2. We then show how to adapt our paradigm to the setting of constrained optimization for a number of important classes of hard problem constraints. For this we compare and contrast both penalty-based and feasibility-preserving approaches, elucidating the significant advantages of the latter approach. Our approach is general and may be applied to easy-to-prepare initial states as a standalone algorithm, or deployed as a quantum postprocessing stage to improve performance of a given parameterized quantum circuit. We then propose a more sophisticated variant of our algorithm that adaptively applies a mixing operator or not, based on the measurement outcomes seen so far, as to speeds up the algorithm and helps the system evolution avoid slowing down or getting stuck suboptimally. In particular, we show that mixing operators from the quantum alternating operator ansatz can be imported directly, both for the necessary eigenstate scrambling operator and for initial state preparation, and discuss quantum resource tradeoffs.
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).
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).