arXiv subjects

Boris Arseniev

Publications and source records attributed to Boris Arseniev.

3 recordsLinked to original sources

arXiv2024

High order schemes for solving partial differential equations on a quantum computer

We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing $d$-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.

quant-ph↗

arXiv2024

Distributed quantum logic algorithm

Parallel computation enables multiple processors to execute different parts of a task simultaneously, improving processing speed and efficiency. In quantum computing, parallel gate implementation involves executing gates independently in different registers, directly impacting the circuit depth, the number of sequential quantum gate operations, and thus the algorithm execution time. This work examines a method for reducing circuit depth by introducing auxiliary qubits to enable parallel gate execution, potentially enhancing the performance of quantum simulations on near-term quantum devices. We show that any circuit on $n$ qubits with depth $O\left(M n^2\right)$, where $M = M(n)$ is some function of $n$, can be transformed into a circuit with depth $O\left(\log_2(M) n^2\right)$ operating on $O\left(M n\right)$ qubits. This technique may be particularly useful in noisy environments, where recent findings indicate that only the final $O\left(\log n\right)$ layers influence the expectation value of observables. It may also optimize Trotterization by exponentially reducing the number of Trotter steps. Additionally, the method may offer advantages for distributed quantum computing, and the intuition of treating quantum states as gates and operators as vectors used in this work may have broader applications in quantum computation.

quant-ph↗

arXiv2024

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix $2^n \times 2^n$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as function of the number of qubits is lower than the oracle based approach for $n < 15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.

quant-ph↗