In this work, we investigate the trade-off between the circuit depth and the number of ancillary qubits for preparing sparse quantum states. We prove that any $n$-qubit $d$-spare quantum state (i.e., it has only $d$ non-zero amplitudes) can be prepared by a quantum circuit with depth $O\left(\frac{nd \log m}{m \log m/n} + \log nd\right)$ using $m\geq 6n$ ancillary qubits, which achieves the current best trade-off between depth and ancilla number. In particular, when $m = Θ({\frac{nd}{\log d}})$, our result recovers the optimal circuit depth $Θ(\log nd)$ given in \hyperlink{cite.zhang2022quantum}{[Phys. Rev. Lett., 129, 230504(2022)]}, but using significantly fewer gates and ancillary qubits.
In this work, we investigate the trade-off between the circuit depth and the number of ancillary qubits for preparing sparse quantum states. We prove that any n-qubit d-spare quantum state (i.e., it has only d non-zero amplitudes) can be prepared by a quantum circuit with depth O(mlogm/nndlogm+lognd) using m≥6n ancillary qubits, which achieves the current best trade-off between depth and ancilla number. In particular, when m=Θ(logdnd), our result recovers the optimal circuit depth Θ(lognd) given in \hyperlink{cite.zhang2022quantum}{[Phys. Rev. Lett., 129, 230504(2022)]}, but using significantly fewer gates and ancillary qubits.
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).
The Quantum State Preparation problem aims to prepare an n-qubit quantum state |ψv=k=02n-1vk|k from the initial state |0n, for a given unit vector v=(v0,v1,v2,v2n-1)TC2n with ||v||2=1. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open when ancillary qubits are available. In this paper, we study quantum circuits when there are m ancillary qubits available. We construct, for any m, circuits that can prepare |ψvin depth Õ(2nm+n+n) and size O(2n), achieving the optimal value for both measures simultaneously. These results also imply a depth complexity of (4nm+n) for quantum circuits implementing a general n-qubit unitary for any m≤O(2n/n) number of ancillary qubits. This resolves the depth complexity for circuits without ancillary qubits. And for circuits with exponentially many ancillary qubits, our result quadratically improves the currently best upper bound of O(4n) to ˜(2n). Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide parameter regime. The results can be viewed as (optimal) time-space trade-off bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime.
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.
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.
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.
Block encodings are a basic interface between quantum algorithms and linear algebra. Standard LCU constructions achieve optimal circuit depth but typically require logarithmically many ancilla qubits. We ask how much quantum workspace can be reduced without sacrificing circuit depth, and study this tradeoff from both algorithmic and lower-bound perspectives.
For a Hermitian decomposition A=∑j=1LαjHj, with ∥Hj∥=1 and α=∑j∣αj∣, we give two coherent ε-approximate block-encoding constructions. The first uses one ancilla qubit and has depth O(L(α/ε)o(1)), while the second uses O(loglog(α/ε)) ancillas and achieves depth O(L).
For a broad Suzuki-based coherent simulation architecture, we prove an ancilla-depth tradeoff. In the polynomial-resource regime and for a constant number of coherent rounds, log(1/ε)≤O((logQ)2+2alogQ), where Q is depth normalized by the number of Hamiltonian terms and a is the ancilla count. Thus polylogarithmic dependence on 1/ε requires more than constantly many ancillas within this architecture. In a separate repeated-query LCU model, for balanced coefficients 1/L and error ε=η/L with fixed 0<η<1, we prove 2a=Ωη(L2/(T+L)), where T is the number of oracle queries. Hence a=Ω(logL) when T=O(Lα) for some α<2. Moreover, in the exact case, a≥logL regardless of T. We also extend this tradeoff to arbitrary nonnegative coefficients.
Finally, we apply our low-ancilla constructions to normalized trace estimation in DQC1, obtaining an optimal algorithm linear in the approximate degree together with a matching query lower bound. Together, these results establish quantitative space-depth and space-query tradeoffs in two natural circuit models.