We propose a universal quantum circuit design that can estimate any arbitrary one-dimensional periodic functions based on the corresponding Fourier expansion. The quantum circuit contains N -qubits to store the information on the different N -Fourier components and M + 2 auxiliary qubits with M = ⌈log 2 N ⌉ for control operations. The desired output will be measured in the last qubit q N with a time complexity of the computation of , which leads to polynomial speedup under certain circumstances. We illustrate the approach by constructing the quantum circuit for the square wave function with accurate results obtained by direct simulations using the IBM-QASM simulator. The approach is general and can be applied to any arbitrary periodic function.
We propose a universal quantum circuit design that can estimate any arbitrary one-dimensional periodic functions based on the corresponding Fourier expansion. The quantum circuit contains N -qubits to store the information on the different N -Fourier components and M + 2 auxiliary qubits with M = ⌈log 2 N ⌉ for control operations. The desired output will be measured in the last qubit q N with a time complexity of the computation of , which leads to polynomial speedup under certain circumstances. We illustrate the approach by constructing the quantum circuit for the square wave function with accurate results obtained by direct simulations using the IBM-QASM simulator. The approach is general and can be applied to any arbitrary periodic function.
Abstract The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms, including those for solving partial differential equations and Monte Carlo estimation. In this work, we introduce the Fourier series loader (FSL) method for preparing quantum states that exactly encode multi-dimensional Fourier series using linear-depth quantum circuits. Specifically, the FSL method prepares a (Dn)-qubit state encoding the 2 Dn -point uniform discretization of a D-dimensional function specified by a D-dimensional Fourier series. A free parameter, m, which must be less than n, determines the number of Fourier coefficients, 2 D ( m + 1 ) , used to represent the function. The FSL method uses a quantum circuit of depth at most 2 ( n − 2 ) + ⌈ log 2 ( n − m ) ⌉ + 2 D ( m + 1 ) + 2 − 2 D ( m + 1 ) , which is linear in the number of Fourier coefficients, and linear in the number of qubits (Dn) despite the fact that the loaded function’s discretization is over exponentially many (2 Dn ) points. The FSL circuit consists of at most D n + 2 D ( m + 1 ) + 1 − 1 single-qubit and D n ( n + 1 ) / 2 + 2 D ( m + 1 ) + 1 − 3 D ( m + 1 ) − 2 two-qubit gates; we present a classical compilation algorithm with runtime O ( 2 3 D ( m + 1 ) ) to determine the FSL circuit for a given Fourier series. The FSL method allows for the highly accurate loading of complex-valued functions that are well-approximated by a Fourier series with finitely many terms. We report results from noiseless quantum circuit simulations, illustrating the capability of the FSL method to load various continuous 1D functions, and a discontinuous 1D function, on 20 qubits with infidelities of less than 10−6 and 10−3, respectively. We also demonstrate the practicality of the FSL method for near-term quantum computers by presenting experiments performed on the Quantinuum H1-1 and H1-2 trapped-ion quantum computers: we loaded a complex-valued function on 3 qubits with a fidelity of over 95 % , as well as various 1D real-valued functions on up
A single-qubit circuit can approximate any bounded complex function stored in the degrees of freedom defining its quantum gates. The single-qubit approximant presented in this work is operated through a series of gates that take as their parametrization the independent variable of the target function and an additional set of adjustable parameters. The independent variable is re-uploaded in every gate while the parameters are optimized for each target function. The output state of this quantum circuit becomes more accurate as the number of re-uploadings of the independent variable increases, i.e., as more layers of gates parameterized with the independent variable are applied. In this work, we provide two proofs of this claim related to both the Fourier series and the universal approximation theorem for neural networks, and we benchmark both methods against their classical counterparts. We further implement a single-qubit approximant in a real superconducting qubit device, demonstrating how the ability to describe a set of functions improves with the depth of the quantum circuit. This work shows the robustness of the re-uploading technique on quantum machine learning.
What makes a class of quantum circuits efficiently classically simulable on average? I present a framework that applies harmonic analysis of groups to circuits with a structure encoded by group parameters. Expanding the circuits in a suitable truncated multi-path operator basis gives algorithms to evaluate the Fourier coefficients of output distributions or expectation values that are viewed as functions on the group. Under certain conditions, a truncated Fourier series can be efficiently estimated with guaranteed mean-square convergence. For classes of noisy circuits, it leads to algorithms for sampling and mean value estimation under error models with a spectral gap, where the complexity increases exponentially with the gap's inverse and polynomially with the circuit's size. This approach unifies and extends existing algorithms for noisy parametrised or random circuits using Pauli basis paths. For classes of noiseless circuits, mean values satisfying Lipschitz continuity can be on average approximated using efficient sparse Fourier decompositions. I also discuss generalisations to homogeneous spaces, qudit systems and a way to analyse random circuits via matrix coefficients of irreducible representations.
We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits. To execute an arbitrary quantum circuit on n qubits, we give a three-dimensional quantum fault-tolerance architecture involving O ( n 3 / 2 log 3 n ) qubits and a quasi-two-dimensional architecture using O ( n 2 log 3 n ) qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance-threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead, and a quasipolylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing.
We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary inverse polynomial in n. This improves the method of A.Y. Kitaev (1995) which requires time quadratic in n. This algorithm also leads to a general and efficient Fourier sampling technique which improves upon the quantum Fourier sampling lemma of L. Hales and S. Hallgren (1997). As an application of this technique, we give a quantum algorithm which finds the period of an arbitrary periodic function, i.e. a function which may be many-to-one within each period. We show that this algorithm is efficient (polylogarithmic in the period of the function) for a large class of periodic functions. Moreover, using standard quantum lower-bound techniques, we show that this characterization is right. That is, this is the maximal class of periodic functions with an efficient quantum period-finding algorithm.
Unlike fixed designs, programmable circuit designs support an infinite number of operators. The functionality of a programmable circuit can be altered by simply changing the angle values of the rotation gates in the circuit. Here, we present a new quantum circuit design technique resulting in two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix-which can be non-unitary-in an efficient way. We also give both the classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator U = e(-iHt) for a given atomic and molecular Hamiltonian H in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. Since the presented circuit designs are independent from the matrix decomposition techniques and the global optimization processes used to find quantum circuits for a given operator, high accuracy simulations can be done for the unitary propagators of molecular Hamiltonians on quantum computers. As an example, we show how to build the circuit design for the hydrogen molecule.