Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications

Zhang, Xiaoming; Li, Tongyang; Yuan, Xiao

doi:10.48550/arxiv.2201.11495

Cited by 17 publications

(20 citation statements)

References 35 publications

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“…where N is the normalization factor, N = 2 n , |j is a standard basis vector with j ∈ {0, 1, ..., N − 1}, and x j is the j th grid point in a uniform discretization of the interval [a, b]-while we later remark that non-uniform grid spacings may sometimes be advantageous. In general, preparing an arbitrary state in a Hilbert space of n qubits requires exponential complexity as the dimension of the space grows exponentially in n [17,20]. However, the algorithm we are about to present demonstrates that the very general set of states of the form shown in Eq.…”

Section: Algorithm Overviewmentioning

confidence: 99%

“…Unfortunately, a quantum circuit capable of producing an arbitrary quantum state necessitates exponential complexity in general [17], although recent works have shown that the exponential circuit depth cost can actually be made linear in exchange for utilizing an exponential number of ancillary qubits [18][19][20]. As a result, in practice, it is essential to exploit specific properties of the state being produced.…”

Section: Introductionmentioning

confidence: 99%

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Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity

Rattew¹,

Koczor²

2022

Preprint

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Quantum computers will be able solve important problems with significant polynomial and exponential speedups over their classical counterparts, for instance in option pricing in finance, and in real-space molecular chemistry simulations. However, key applications can only achieve their potential speedup if their inputs are prepared efficiently. We effectively solve the important problem of efficiently preparing quantum states following arbitrary continuous (as well as more general) functions with complexity logarithmic in the desired resolution, and with rigorous error bounds. This is enabled by the development of a fundamental subroutine based off of the simulation of rank-1 projectors. Combined with diverse techniques from quantum information processing, this subroutine enables us to present a broad set of tools for solving practical tasks, such as state preparation, numerical integration of Lipschitz continuous functions, and superior sampling from probability density functions. As a result, our work has significant implications in a wide range of applications, for instance in financial forecasting, and in quantum simulation.

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Section: Algorithm Overviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity

Rattew¹,

Koczor²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Later, Rosenthal independently constructed a QSP circuit of depth O(n) using O(n2 n ) ancillary qubits [Ros21]. This year, [ZLY22] gave yet another proof of the O(n) depth upper bound using O(2 n ) ancillary qubits. Both [Ros21] and [ZLY22] did not give results for general m.…”

Section: Introductionmentioning

confidence: 99%

“…This year, [ZLY22] gave yet another proof of the O(n) depth upper bound using O(2 n ) ancillary qubits. Both [Ros21] and [ZLY22] did not give results for general m.…”

Section: Introductionmentioning

confidence: 99%

Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits

Pan¹,

Zhang²

2022

Preprint

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As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, QRAM aims to provide the transformation of |i |0 n → |i |ψ i for all i ∈ {0, 1} k for the given nqubit states |ψ i . In this paper, we construct a quantum circuit for implementing QRAM, with depthn+k+m and size O 2 n+k for any given number m of ancillary qubits. These bounds, which can also be viewed as time-space tradeoff for the transformation, are optimal for any integer parameters m, k ≥ 0 and n ≥ 1.Our construction induces a circuit for the standard quantum state preparation (QSP) problem of |0 n → |ψ for any |ψ , pinning down its depth complexity to Θ(n+2 n /(n+m)) and its size complexity to Θ(2 n ) for any m. This improves results in a recent line of investigations of QSP with ancillary qubits.Another fundamental problem, unitary synthesis, asks to implement a general n-qubit unitary by a quantum circuit. Previous work shows a lower bound of Ω(n + 4 n /(n + m)) and an upper bound of O(n2 n ) for m = Ω(2 n /n) ancillary qubits. In this paper, we quadratically shrink this gap by presenting a quantum circuit of depth of O n2 n/2

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“…Here the cost is quantified by the number of required CNOT gates, as any quantum circuit can be decomposed into CNOT gates and single-qubit gates and the number of single-qubit gates is upper bounded by twice the number of CNOTs [13]. In this work, we focus on algorithms that prepare quantum states in a deterministic manner with no or fixed ancillary qubit overhead, instead of approximate algorithms [14][15][16] or algorithms with n-dependent ancilla size [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Efficient Deterministic Preparation of Quantum States Using Decision Diagrams

Mozafari,

De Micheli,

Yang

2022

Preprint

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Loading classical data into quantum registers is one of the most important primitives of quantum computing. While the complexity of preparing a generic quantum state is exponential in the number of qubits, in many practical tasks the state to prepare has a certain structure that allows for faster preparation. In this paper, we consider quantum states that can be efficiently represented by (reduced) decision diagrams, a versatile data structure for the representation and analysis of Boolean functions. We design an algorithm that utilises the structure of decision diagrams to prepare their associated quantum states. Our algorithm has a circuit complexity that is linear in the number of paths in the decision diagram. Numerical experiments show that our algorithm reduces the circuit complexity by up to 31.85% compared to the state-of-the-art algorithm, when preparing generic n-qubit states with different degrees of non-zero amplitudes. Additionally, for states with sparse decision diagrams, including the initial state of the quantum Byzantine agreement protocol, our algorithm reduces the number of CNOTs by 86.61% ∼ 99.9%.

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Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications

Cited by 17 publications

References 35 publications

Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity

Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity

Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits

Efficient Deterministic Preparation of Quantum States Using Decision Diagrams

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