Efficient quantum circuits for diagonal unitaries without ancillas

Abstract

The accurate evaluation of diagonal unitary operators is often the most resourceintensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitaryê ( ) if x in the x basis, to that of finding the minimallength Walsh-series approximation to the function f(x). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 033040 J Welch et al. The second equality shows that the functional form is the same whether x is continuous or discrete, the only difference being the number of bits in the expansion of x. This makes Walsh series useful for representing discretely sampled continuous functions. New J. Phys. 16 (2014) 033040 J Welch et al =Û e if that is diagonal in this basis is given in terms of its eigenvalues asˆ= f k f k k . New J. Phys. 16 (2014) 033040 J Welch et al 0 1 j j New J. Phys. 16 (2014) 033040 J Welch et al 5 New J. Phys. 16 (2014) 033040 J Welch et al 7 7, which means that 7 qubits are necessary to represent the series. New J. Phys. 16 (2014) 033040 J Welch et al

Quantum Computation using Arrays of N Polar Molecules in Pendular States

“…Of course, the belief abouts uch difficulty with classical computers is based on well-known conjectures in computational complexity.H ere we consider ac lass of circuits known as IQP (instantaneousq uantum polynomial time), which was introduced in Refs. [60,61] In particular,w es tress that for any nqubit diagonal unitary D,u sing techniques from Ref. Equivalently one couldr egarda nI QP circuit as as equence of commuting quantum gates.…”

“…[35,36].A nn-qubit IQP circuit takes the form H n DH n where D is adiagonal unitaryoperator. [61] we could construct an approximation circuitD acting on the same n qubitsu sing only nearest neighborC NOT gates (see Section 6). As simple as they seem, the output distributions of IQP circuits are classically hard to sample from in the worst case.…”

“…It is known that any Hamiltonian H for ac ontinuous time quantumw alko na ny circulant graph can be diagonalized by the unitary Fourier Transform: [63] H ¼ Q y LQ where L is diagonal. [60,61] Hence aq uantum walk on ac irculant graph of 2 n nodes is efficiently realizable in the gate model with n qubits.M ore interesting is the prospecto fr eplacing the n-qubitq uantum Fouriert ransform Q with easierto-implement Hadamard gates H n when it comes to computing the probability of obtaining j00 ÁÁÁ0i in the final state [Eq. Quantum Fourier transform Q is well-known to be efficientlyr ealizable,a nd general schemes also exist for simulating diagonal unitaries.…”

Quantum Information and Computation for Chemistry

“…In particular, this work sparked the development of quantum algorithms for the required task of quantum state preparation (Kassal et al, 2008), and similar approaches which handle multi-reference states (Wang et al, 2008). Researchers have also considered extending the use of quantum algorithms to various regimes in quantum chemistry such as relativistic dynamics (Veis et al, 2012) and quantum dynamics beyond the Born-Oppenheimer approximation (Kassal et al, 2008;Welch et al, 2014;. Error correction for quantum algorithms for quantum chemistry has also been investigated (Jones et al, 2012).…”

“…Subsequent works have opened up an entire line of inquiry concerning real-space simulation of quantum chemistry (Kassal et al, 2008;Welch et al, 2014;Kivlichan et al, 2016). In particular, it has been found that for systems with more than roughly 4 atoms, real space simulation is both more efficient and more accurate than second-quantized methods (Kassal et al, 2008;Kivlichan et al, 2016).…”