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.
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.
Quantum state preparation is an important subroutine for quantum computing. We show that any n-qubit quantum state can be prepared with a Θ(n)-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with d⩾2 nonzero entries, we can reduce the circuit depth to Θ(log(nd)) with O(ndlogd) ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.
Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation on the circuit size (the total count of elementary gates in the circuit) for sparse quantum state preparation. A quantum state is said to be d-sparse if it has only d non-zero amplitudes. For the task of preparing an n-qubit d-sparse quantum state, we obtain the following results: \textbf{Without ancillary qubits:} Any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O(lognnd+n) without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when d=poly(n), and this optimality holds for a broader scope under some reasonable assumptions. \textbf{With limited ancillary qubits:} (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O(log(n+m)nd+n) using m ancillary qubits for any m∈O(logndnd+n). (ii) We establish a matching lower bound Ω(log(n+m)nd+n) under some reasonable assumptions, and obtain a slightly weaker lower bound Ω(log(n+m)+logdnd+n) without any assumptions. \textbf{With unlimited ancillary qubits:} Given arbitrary amount of ancillary qubits available, the circuit size for preparing n-qubit d-sparse quantum states is Θ(logndnd+n).
A crucial subroutine in quantum computing is to load the classical data of N complex numbers into the amplitude of a superposed $n=\ensuremath{\lceil}{log}_{2}N\ensuremath{\rceil}$-qubit state. It has been proven that any algorithm universally implementing this subroutine would need at least O(N) constant weight operations. However, the proof assumes that only n qubits are used, whereas the circuit depth could be reduced by extending the space and allowing ancillary qubits. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with O(n2) circuit depth to encode any N complex numbers using only single- and two-qubit gates, and local measurements with ancillary qubits. Different variances of the algorithm are proposed with different space and runtime. In particular, we present a scheme with O(N2) ancillary qubits, O(n2) circuit depth, and O(n2) average runtime, which exponentially improves the conventional bound. While the algorithm requires more ancillary qubits, it consists of quantum circuit blocks that only simultaneously act on a constant number of qubits, and at most O(n) qubits are entangled. We also prove a fundamental lower bound $\mathrm{\ensuremath{\Omega}}(n)$ for the minimum circuit depth and runtime with an arbitrary number of ancillary qubits, aligning with our scheme with O(n2). The algorithms are expected to have wide applications in both near-term and universal quantum computing.
A crucial subroutine in quantum computing is to load the classical data of N complex numbers into the amplitude of a superposed n=⌈log2N⌉-qubit state. It has been proven that any algorithm universally implementing this subroutine would need at least O(N) constant weight operations. However, the proof assumes that only n qubits are used, whereas the circuit depth could be reduced by extending the space and allowing ancillary qubits. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with O(n2) circuit depth to encode any N complex numbers using only single-, two-qubit gates and local measurements with ancillary qubits. Different variances of the algorithm are proposed with different space and runtime. In particular, we present a scheme with O(N2) ancillary qubits, O(n2) circuit depth, and O(n2) average runtime, which exponentially improves the conventional bound. While the algorithm requires more ancillary qubits, it consists of quantum circuit blocks that only simultaneously act on a constant number of qubits and at most O(n) qubits are entangled. We also prove a fundamental lower bound $\mitΩ(n)$ for the minimum circuit depth and runtime with arbitrary number of ancillary qubits, aligning with our scheme with O(n2). The algorithms are expected to have wide applications in both near-term and universal quantum computing.
As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, controlled quantum state preparation (CQSP) aims to provide the transformation of | i ⟩ | 0 n ⟩ → | i ⟩ | ψ i ⟩ for all i ∈ { 0 , 1 } k for the given n -qubit states | ψ i ⟩ . In this paper, we construct a quantum circuit for implementing CQSP, with depth O ( n + k + 2 n + k n + k + m ) and size O ( 2 n + k ) for any given number m of ancillary qubits. These bounds, which can also be viewed as a time-space tradeoff for the transformation, are optimal for any integer parameters m , k ≥ 0 and n ≥ 1 . When k = 0 , the problem becomes the canonical quantum state preparation (QSP) problem with ancillary qubits, which asks for efficient implementations of the transformation | 0 n ⟩ | 0 m ⟩ → | ψ ⟩ | 0 m ⟩ . This problem has many applications with many investigations, yet its circuit complexity remains open. Our construction completely solves this problem, pinning down its depth complexity to Θ ( n + 2 n / ( n + m ) ) and its size complexity to Θ ( 2 n ) for any
The preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number s of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares s-sparse states on n qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a ⌈log(s)⌉-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially 2s for sufficiently large values of n, yielding roughly a log(s)/2 improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to s. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.