arXiv:2304.01704 · Quantum Science and Technology

Efficient parallelization of quantum basis state shift

Ljubomir Budinski, Ossi Niemimäki, Roberto Zamora-Zamora, Valtteri Lahtinen

Published Sep 18, 2023• 12 citations

Abstract

Abstract Basis state shift is central to many quantum algorithms, most notably the quantum walk. Efficient implementations are of major importance for achieving a quantum speedup for computational applications. We optimize the state shift algorithm by incorporating the shift in different directions in parallel. This provides a significant reduction in the depth of the quantum circuit in comparison to the currently known methods, giving a linear scaling in the number of gates versus working qubits in contrast to the quadratic scaling of the state-of-the-art method based on the quantum Fourier transform. For a one-dimensional array of size 2 n for n > 4, we derive the total number of 15 n + 74 two-qubit CX gates in the parallel circuit, using a total of 2 n − 2 qubits including an ancilla register for the decomposition of multi-controlled gates. We focus on the one-dimensional and periodic shift, but note that the method can be extended to more complex cases.

DOI: 10.1088/2058-9565/acfab7

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arXiv:2304.01704 · Quantum Science and Technology

Efficient parallelization of quantum basis state shift

Ljubomir Budinski, Ossi Niemimäki, Roberto Zamora-Zamora, Valtteri Lahtinen

Published Sep 18, 2023• 12 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/2058-9565/acfab7

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The preparation of

n

-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when

m

ancillary qubits are available. The reduction in complexity is due to the use of a single operator

Λ

for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.

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.