This paper addresses the challenge of preparing mixed quantum states -- both arbitrary states in general and highly sparse ones in particular -- an area that has received far less attention than the preparation of pure states. We present two classes of circuit-synthesis methods: one based on constructing the density matrix as a mixture of pure states and the other based on purification. To improve both preprocessing efficiency and the complexity of the resulting circuits, we propose a novel strategy based on the Cholesky decomposition, which offers significant advantages, especially when the target density matrix is highly or extremely sparse. Furthermore, by exploiting incomplete Cholesky decomposition with threshold dropping, we introduce a practical approach for constructing high-fidelity approximations of the target density matrix. This approach enables substantial reductions in circuit complexity at the cost of negligible or only mild fidelity loss.
Preparing a quantum circuit that implements a given sparse state is an important building block that is necessary for many different quantum algorithms. In the context of fault-tolerant quantum computing, the so-called non-Clifford gates are much more expensive to perform than the Clifford ones. We hence provide an algorithm for synthesizing sparse quantum states with a special care for quantum resources. The circuit depth, ancilla count, and crucially non-Clifford count of the circuit produced by the algorithm are all linear in the sparsity when access to arbitrary-angled rotations is given. When compiled down to the standard Clifford+T gate set, several constructions can be given for increasingly better T-count and depth at the expense of a larger number of ancillae. The most optimised construction for T-count reaches
O(s+log2(1/ε)) T gates for error
ε. The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.
Near-term quantum computers have significant error rates and short coherence times, so compilation of circuits to be as short as possible is essential. Two types of compilation problems are typically considered: circuits to prepare a given state from a fixed input state, called “state preparation”; and circuits to implement a given unitary operation, for example by “unitary synthesis”. In this paper we solve a more general problem: the transformation of a set of m states to another set of
mstates, which we call “multi-state preparation”. State preparation and unitary synthesis are special cases; for state preparation, m=1, while for unitary synthesis, m is the dimension of the full Hilbert space. We generate and optimize circuits for multi-state preparation numerically. In cases where a top-down approach based on matrix decompositions is also possible, our method finds circuits with substantially (up to 40 %) fewer two-qubit gates. We discuss possible applications, including efficient preparation of macroscopic superposition (“cat”) states and synthesis of quantum channels.
For practical usage of quantum systems in solving real-world problems, large amounts of classical data are required to be transferred/encoded to the quantum domain. Arbitrary classical data is usually encoded onto quantum devices by synthesizing and initializing a corresponding quantum state. Current techniques of arbitrary state synthesis, however, produce deep and complex quantum circuits, leading to low state fidelity and possible violations of decoherence constraints. In this work, we propose an improved methodology and optimized circuits for synthesizing any arbitrary quantum state from given classical data. Compared to existing methods, the proposed methodology results in circuits with lower gate count, lower circuit depth, and high state fidelity. The proposed methods are evaluated by simulation in MATLAB and IBM qasm, and realistic implementation on an IBM quantum device. The experimental results show a reduction in gate count and circuit depth by a factor of two over existing methods.
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.
Future quantum information processing devices will require the use of exotic quantum states, such as specially crafted entangled states, to achieve certain desired computations on demand. Thus far, synthesis schemes for such states have been devised on a case-by-case basis using ad hoc techniques. In this paper we present a systematic method for finding a quantum circuit that can synthesize any pure or mixed n-qubit state. We then give examples of the use of our algorithm for finding synthesis pathways for especially exotic quantum states such as maximal mixed states. It is not known how to prepare general instances of such states by other means. Thus our quantum state synthesis algorithm should be of use not only in quantum information processing, but also in experimental quantum physics.
We consider the task of breaking down a quantum computation given as an isometry into C-nots and single-qubit gates, while keeping the number of C-not gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.