Quantum state preparation is a fundamental task in quantum information science with wide applications in computing, communication, and precision measurement. This work proposes three quantum state preparation schemes based on quantum walk with position–time-dependent coin operators. Scheme 1 enables the preparation of arbitrary high-dimensional quantum states, while scheme 2 targets a specific class of states with significantly lower resource costs. Building on these, we propose the scheme 3 that combines the universality of scheme 1 and the efficiency of scheme 2, forming the core contribution of this paper. We demonstrate the effectiveness of these schemes through examples and quantum circuit implementations. The coin operations can be realized with depth <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(\frac{2^n}{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mfrac> <msup> <mn>2</mn> <mi>n</mi> </msup> <mi>n</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and size <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(2^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which can be further reduced to O(kn) when only a few positions require nontrivial operations. Shift operations can be implemented with constant depth. In general, the circuit complexity for all three schemes is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$O(2^{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. However, for spare states, scheme 2 and scheme 3 can reduce the circuit complexity to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(kn^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This framework provides an efficient and scalable approach to quantum state preparation, with potential applications in quantum algorithms and beyond.

Efficient preparation method for arbitrary multiqubit states based on quantum walk

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发表日期：

2025年10月21日

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Dan Li,
Lei Zhong

摘要

Quantum state preparation is a fundamental task in quantum information science with wide applications in computing, communication, and precision measurement. This work proposes three quantum state preparation schemes based on quantum walk with position–time-dependent coin operators. Scheme 1 enables the preparation of arbitrary high-dimensional quantum states, while scheme 2 targets a specific class of states with significantly lower resource costs. Building on these, we propose the scheme 3 that combines the universality of scheme 1 and the efficiency of scheme 2, forming the core contribution of this paper. We demonstrate the effectiveness of these schemes through examples and quantum circuit implementations. The coin operations can be realized with depth $O(\frac{2^n}{n})$ $O (\frac{2^{n}}{n})$ and size $O(2^n)$ $O (2^{n})$ , which can be further reduced to O(kn) when only a few positions require nontrivial operations. Shift operations can be implemented with constant depth. In general, the circuit complexity for all three schemes is $O(2^{n})$ $O (2^{n})$ . However, for spare states, scheme 2 and scheme 3 can reduce the circuit complexity to $O(kn^{2})$ $O (k n^{2})$ . This framework provides an efficient and scalable approach to quantum state preparation, with potential applications in quantum algorithms and beyond.