Quantum phase estimation is a corner-stone in quantum algorithm design, al-lowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these... Show moreQuantum phase estimation is a corner-stone in quantum algorithm design, al-lowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, -known as the Heisen-berg limit-, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estima-tion that do not require coherence between experiments have garnered interest in re-cent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is un-able to prepare eigenstates of the system. Given a quantum subroutine which pro-vides samples of a 'phase function' g(k) = sigma(j) A(j)e(i Phi)j(k) with unknown eigenphases phi(j )and overlaps A(j )at quantum cost O(k), we show how to estimate the phases {phi(j}) with (root-mean-square) error delta for total quantum cost T = O(delta(-1)). Our scheme com-bines the idea of Heisenberg-limited multi -order quantum phase estimation for a single eigenvalue phase [1, 2] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [3] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberg -limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well. Show less
Modeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular... Show moreModeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0.0140.014 Å of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H22 molecule, finding agreement at the equilibrium bond length to within 0.060.06 a.u. (2%2% relative error). Show less
Sagastizabal, R.; Bonet Monroig, X.; Singh, M.; Rol, M.A.; Bultink, C.C.; Fu, X.; ... ; DiCarlo, L. 2019
Variational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by... Show moreVariational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by applying the recently proposed symmetry verification technique to the experimental estimation of the ground-state energy and ground state of the hydrogen molecule. A finely adjustable exchange interaction between two qubits in a circuit QED processor efficiently prepares variational ansatz states in the single-excitation subspace respecting the parity symmetry of the qubit-mapped Hamiltonian. Symmetry verification improves the energy and state estimates by mitigating the effects of qubit relaxation and residual qubit excitation, which violate the symmetry. A full-density-matrix simulation matching the experiment dissects the contribution of these mechanisms from other calibrated error sources. Enforcing positivity of the measured density matrix via scalable convex optimization correlates the energy and state estimate improvements when using symmetry verification, with interesting implications for determining system properties beyond the ground-state energy. Show less
This thesis covers various applications of topology in condensed matter physics and quantum information. It studies how the topology of the electronic structure of a Weyl semimetal affects the... Show moreThis thesis covers various applications of topology in condensed matter physics and quantum information. It studies how the topology of the electronic structure of a Weyl semimetal affects the transport behaviour of electrons in an applied magnetic field, and how one may employ similar ideas in materials containing Majorana modes to speed up chemistry calculations on a quantum computer. It develops and tests new techniques for decoding topological quantum error correcting codes, in particular for detailed simulation on near-term devices. Finally, it looks towards improving quantum algorithms for future applications in quantum simulation; in particular the classical post-processing of data taken during quantum phase estimation experiments. Show less
A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the... Show moreA quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment—without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate L of the encoded logical qubit to values much below the error rate phys of the physical qubits—fitting the expected power law scaling , with d the code distance. The neural network incorporates the information from 'flag qubits' to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters. Show less
Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE... Show moreQuantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H-4-molecule. Show less
Quantum error correction of a surface code or repetition code requires the pairwise matching of error events in a space-time graph of qubit measurements, such that the total weight of the matching... Show moreQuantum error correction of a surface code or repetition code requires the pairwise matching of error events in a space-time graph of qubit measurements, such that the total weight of the matching is minimized. The input weights follow from a physical model of the error processes that affect the qubits. This approach becomes problematic if the system has sources of error that change over time. Here, it is shown that the weights can be determined from the measured data in the absence of an error model. The resulting adaptive decoder performs well in a time-dependent environment, provided that the characteristic timescale tau(env) of the variations is greater than delta t/(p) over bar, with dt the duration of one error-correction cycle and (p) over bar the typical error probability per qubit in one cycle. Show less
Bonet-Monroig, X.; Sagastizabal, R.; Singh, M.; O'Brien, T.E. 2018
We investigate the performance of error mitigation via measurement of conserved symmetries on near-term devices. We present two protocols to measure conserved symmetries during the bulk of an... Show moreWe investigate the performance of error mitigation via measurement of conserved symmetries on near-term devices. We present two protocols to measure conserved symmetries during the bulk of an experiment, and develop a third, zero-cost, post-processing protocol which is equivalent to a variant of the quantum subspace expansion. We develop methods for inserting global and local symmetries into quantum algorithms, and for adjusting natural symmetries of the problem to boost the mitigation of errors produced by different noise channels. We demonstrate these techniques on two- and four-qubit simulations of the hydrogen molecule (using a classical density-matrix simulator), finding up to an order of magnitude reduction of the error in obtaining the ground-state dissociation curve. Show less
