arXiv subjects

Alexander I. Zenchuk

Publications and source records attributed to Alexander I. Zenchuk.

9 recordsLinked to original sources

arXiv2026

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

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.

quant-ph↗

arXiv2026

Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues

We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary $N\times N$ 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, $O(N^2 \log N)$ and $O(\log N)$.

quant-ph↗

arXiv2025

Matrix encoding method in variational quantum singular value decomposition

We propose the variational quantum singular value decomposition based on encoding the elements of the considered { $N\times N$} 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/\varepsilon$, where $\varepsilon$ is the precision required for calculation of singular values.

quant-ph↗

arXiv2025

Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems

We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an $(N-1)\times (N-1)$ matrix (depth is $O(N^2\log N)$) which is a simple modification of the algorithm for calculating the determinant of an $N\times N$ 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 $\sim 2^{-O(N\log N)}$) 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).

quant-ph↗

arXiv2025

Arbitrary state creation via controlled measurement

The initial state creation is a starting point of many quantum algorithms and usually is considered as a separate subroutine not included into the algorithm itself. There are many algorithms aimed on creation of special class of states. Our algorithm allows creating 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. However, the crucial operation is the final controlled measurement of the ancilla state that removes the garbage part of the superposition state and allows to avoid the problem of small success probability in that measurement. We emphasize that rotation angles are predicted in advance by the required precision and therefore there is no classical calculation supplementing quantum algorithm. The depth and space of the algorithm growth with $n$ as, respectively, $O(2^n n)$ and $O(n)$. This algorithm can be a subroutine generating the required input state in various algorithms, in particular, in matrix-manipulation algorithms developed earlier.

quant-ph↗

arXiv2025

Remarks on controlled measurement and quantum algorithm for calculating Hermitian conjugate

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.

quant-ph↗

arXiv2024

Matrix manipulations via unitary transformations and ancilla-state measurements

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 the addition protocol is $O(1)$ and that of other protocols logarithmically increases with the dimensionality of the considered matrices.

quant-ph↗

arXiv2024

Some Aspects of Remote State Restoring in State Transfer Governed by XXZ-Hamiltonian

We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence of the Larmor frequencies. Restoring in spin chains with up to 20 nodes is studied. Studying PTZ, we consider the zigzag and rectangular configurations and optimize the transfer of the 0-order coherence matrix using geometrical parameters of the communication line as well as the special unitary transformation of the extended receiver. Overall observation is that XXZ-chains require longer time for state transfer than XX-chains, which is confirmed by the analytical study of the evolution under the nearest-neighbor approximation. We demonstrate the exponential increase of the state-transfer time with the spin chain length.

quant-ph↗

arXiv2010

Multiple quantum NMR of spin-carrying molecules in nanopores: high order corrections to the two-spin/two-quantum Hamiltonian

This paper is devoted to the multiple-quantum (MQ) NMR spectroscopy in nanopores filled by a gas of spin-carrying molecules (s=1/2) in a strong external magnetic field. It turned out that the high symmetry of the spin system in nanopores yields a possibility to overcome the problem of the exponential growth of the Hilbert space dimension with an increase in the number of spins and to investigate MQ NMR dynamics in systems consisting of several hundred spins. We investigate the dependence of the MQ coherence intensities on their order (the profile of the MQ coherence intensities) for a spin system governed by the standard MQ NMR Hamiltonian (the nonsecular two-quantum/two-spin Hamiltonian) together with the second order correction of the average Hamiltonian theory. It is shown that the profile depends on the value of this correction and varies from the exponential to the logarithmic one.

physics.chem-ph↗