State preparation is a fundamental routine in quantum computation, for which many algorithms have been proposed. Among them, perhaps the simplest one is the Grover-Rudolph algorithm. In this paper, we analyse the performance of this algorithm when the state to prepare is sparse. We show that the gate complexity is linear in the number of non-zero amplitudes in the state and quadratic in the number of qubits. We then introduce a simple modification of the algorithm, which makes the dependence on the number of qubits also linear. This is competitive with the best known algorithms for sparse state preparation
State preparation is a fundamental routine in quantum computation, for which many algorithms have been proposed. Among them, perhaps the simplest one is the Grover-Rudolph algorithm. In this paper, we analyse the performance of this algorithm when the state to prepare is sparse. We show that the gate complexity is linear in the number of non-zero amplitudes in the state and quadratic in the number of qubits. We then introduce a simple modification of the algorithm, which makes the dependence on the number of qubits also linear. This is competitive with the best known algorithms for sparse state preparation
State preparation is a fundamental routine in quantum computation, for which many algorithms have been proposed. Among them, perhaps the simplest one is the Grover-Rudolph algorithm. In this paper we analyze the performance of this algorithm when the state to prepare is sparse. We show that the gate complexity is linear in the number of nonzero amplitudes in the state and quadratic in the number of qubits. We then introduce a simple modification of the algorithm, which makes the dependence on the number of qubits also linear. This is competitive with the best known algorithms for sparse state preparation.
Sparse quantum state preparation is a common subroutine in quantum algorithms, where classical data with few nonzero entries must be loaded into a quantum state. In this work, we consider the Grover-Rudolph algorithm, which has recently been shown to efficiently prepare sparse states, and we propose two improvements. First, we extend an existing gate-merging procedure by allowing rotations to merge with virtual zero-angle gates on unreachable branches of the preparation tree, reducing the number of CNOTs and control qubits. Second, we introduce an approximate variant in which rotations with similar but not identical angles are merged at the cost of a small, controllable error in the prepared state. We derive a classically computable estimate of the resulting overlap with the target state, which is used to guide the merging decisions.
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.
In order to use a quantum device to assess a classical dataset D, we need to representthe set D in a quantum state. Applying a quantum algorithm, that is a quantum state preparation algorithm, to convert classical data into quantum data would be the common method. Loading classical data into a quantum device is required in many current applications. Efficiently preparing a quantum state to be used as the initial state of a quantum algorithm is an essential step in developing efficient quantum algorithms, since many algorithms need to reload the initial state several times during their execution. The cost to initialize a quantum state can compromise the algorithm efficiency if the process of quantum states preparation is not efficient. The topic of quantum states preparation in quantum computing has been the focus of much attention. In this scope, preparing sparse quantum states is a more specific problem that remains open since many quantum algorithms also require sparse initialization. This dissertation presents the results of an investigation on sparse quantum states preparation with the development of three algorithms, with highlight to the preparation of sparse quantum states, the main contributionof this dissertation. From a classical input dataset with M patterns formed by pairs composed of a complex number and a binary pattern with n bits, this algorithm can prepare a quantum state with n qubits and continuous amplitudes. The cost of its steps is O(nM), classical cost of o(MlogM+nM)and requires a lower CNOT number than the main quantum state preparation algorithms currently known. The preparation of a quantumstate with 2 non-zero amplitudes reveals the need of fewer CNOT gates in n>>1 relation to the main known state preparation algorithms, with even more favorable results with s higher and less 1S in the binary string.
Generating quantum circuits that prepare specific states is an essential part of quantum compilation. Algorithms that solve this problem for general states generate circuits at grow exponentially in the number of qubits. However, in contrast to general states, many practically relevant states are sparse in the standard basis. In this paper we show how sparsity can be used for efficient state preparation. We present a polynomial-time algorithm that generates polynomial-size quantum circuits (linear in the number of nonzero coefficients times number of qubits) that prepare given states, making computer-aided design of sparse state preparation scalable.
We study the fault-tolerant cost of preparing sparse quantum states, measured by T-count in the Clifford+T model. Here an n-qubit state is called s-sparse if it is supported on at most s computational-basis states. For arbitrary n-qubit states, the optimal T-count is Θ(2nlog(1/ε)+log(1/ε)), but for s-sparse states the best previous upper bounds remained linear in s. We show that any n-qubit s-sparse state can be prepared up to error ε using O(min{s,n3/4s}+slog(1/ε)+log(1/ε))T gates, giving the first sublinear dependence on s once the support is sufficiently large. Our approach is based on a support-aware synthesis theorem for sparse Boolean functions, which may be of independent interest. We also prove that, for every 0<ε≤1/6 and 2≤s≤2n/2, sparse-state preparation requires Ω(min{s,ns})T gates, showing that linear dependence on s is unavoidable in the small-support regime and substantially narrowing the gap between the known upper and lower bounds within this parameter range.