The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for
The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediatescale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multiqubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and timereversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the
 Award ID(s):
 1839136
 NSFPAR ID:
 10154197
 Publisher / Repository:
 Nature Publishing Group
 Date Published:
 Journal Name:
 npj Quantum Information
 Volume:
 6
 Issue:
 1
 ISSN:
 20566387
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract N molecular orbitals, the gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a twostep lowrank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with$${\mathcal{O}}({N}^{4})$$ $O\left({N}^{4}\right)$ gate complexity in small simulations, which reduces to$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with$${\mathcal{O}}({N}^{2})$$ $O\left({N}^{2}\right)$ gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ depth on a linearly connected array, an improvement over the$${\mathcal{O}}({N}^{2})$$ $O\left({N}^{2}\right)$ scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearestneighbor twoqubit gates, consisting of fewer than 10^{5}nonClifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ 
Variational approaches are among the most powerful techniques toapproximately solve quantum manybody problems. These encompass bothvariational states based on tensor or neural networks, and parameterizedquantum circuits in variational quantum eigensolvers. However,selfconsistent evaluation of the quality of variational wavefunctionsis a notoriously hard task. Using a recently developed Hamiltonianreconstruction method, we propose a multifaceted approach to evaluatingthe quality of neuralnetwork based wavefunctions. Specifically, weconsider convolutional neural network (CNN) and restricted Boltzmannmachine (RBM) states trained on a square latticespin
1/2 $1/2$ Heisenberg model. We find that the reconstructed Hamiltonians aretypically less frustrated, and have easyaxis anisotropy near the highfrustration point. In addition, the reconstructed Hamiltonians suppressquantum fluctuations in the largeJ_1\!\!J_2 ${J}_{1}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}{J}_{2}$ limit. Our results highlight the critical importance of thewavefunction’s symmetry. Moreover, the multifaceted insight from theHamiltonian reconstruction reveals that a variational wave function canfail to capture the true ground state through suppression of quantumfluctuations.J_2 ${J}_{2}$ 
Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gatelevel error with probability close to one. We model noise by adding a pair of weak, unital, singlequbit noise channels after each twoqubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution
and the corresponding noiseless output distribution$$p_{\text {noisy}}$$ ${p}_{\text{noisy}}$ shrink exponentially with the expected number of gatelevel errors. Specifically, the linear crossentropy benchmark$$p_{\text {ideal}}$$ ${p}_{\text{ideal}}$F that measures this correlation behaves as , where$$F=\text {exp}(2s\epsilon \pm O(s\epsilon ^2))$$ $F=\text{exp}(2s\u03f5\pm O\left(s{\u03f5}^{2}\right))$ is the probability of error per circuit location and$$\epsilon $$ $\u03f5$s is the number of twoqubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution and the uniform distribution$$p_{\text {noisy}}$$ ${p}_{\text{noisy}}$ decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {unif}}$$ ${p}_{\text{unif}}$ . In other words, although at least one local error occurs with probability$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1F)p_{\text {unif}}$$ ${p}_{\text{noisy}}\approx F{p}_{\text{ideal}}+(1F){p}_{\text{unif}}$ , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$1F$$ $1F$ . Thus, the “whitenoise approximation” is meaningful when$$O(F\epsilon \sqrt{s})$$ $O\left(F\u03f5\sqrt{s}\right)$ , a quadratically weaker condition than the$$\epsilon \sqrt{s} \ll 1$$ $\u03f5\sqrt{s}\ll 1$ requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$\epsilon s\ll 1$$ $\u03f5s\ll 1$ , which corresponds to only$$s \ge \Omega (n\log (n))$$ $s\ge \Omega (nlog(n\left)\right)$logarithmic depth circuits, and if, additionally, the inverse error rate satisfies , which is needed to ensure errors are scrambled faster than$$\epsilon ^{1} \ge {\tilde{\Omega }}(n)$$ ${\u03f5}^{1}\ge \stackrel{~}{\Omega}\left(n\right)$F decays. The whitenoise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexitytheoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from secondmoment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds. 
Abstract Twodimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electronspin, is removed by a small Zeeman spin splitting,
, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a reordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$g \mu _B B$$ $g{\mu}_{B}B$ for the$$\nu$$ $\nu $ minimum, e.g., from$$R_{xx}$$ ${R}_{\mathrm{xx}}$ to$$\nu = 11/7$$ $\nu =11/7$ , and a corresponding change in the$$\nu = 8/5$$ $\nu =8/5$ , e.g., from$$R_{xy}$$ ${R}_{\mathrm{xy}}$ to$$R_{xy}/R_{K} = (11/7)^{1}$$ ${R}_{\mathrm{xy}}/{R}_{K}={(11/7)}^{1}$ , with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$R_{xy}/R_{K} = (8/5)^{1}$$ ${R}_{\mathrm{xy}}/{R}_{K}={(8/5)}^{1}$ and$$\nu = 4/3$$ $\nu =4/3$ resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$\nu = 7/5$$ $\nu =7/5$ at the$$R_{xy}$$ ${R}_{\mathrm{xy}}$ minima the latter occurring for$$R_{xx}$$ ${R}_{\mathrm{xx}}$ and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu = 4/3, 7/5$$ $\nu =4/3,7/5$ and$$\nu$$ $\nu $ , but also the tilt or Zeemanenergydependent crossover between distinct FQHE associated with different Hall resistances.$$R_{xy}$$ ${R}_{\mathrm{xy}}$ 
Abstract We study twoqubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlledphase gate CS = diag(1, 1, 1,
i ). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented faulttolerantly in most errorcorrecting schemes through magic state distillation. Since nonClifford gates are typically more expensive to perform in a faulttolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for twoqubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operatorU and outputs a Clifford+CS circuit forU , which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worstcase lower bound of on the number of CS gates required to$$5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)$$ $5{\mathrm{log}}_{2}\left(\frac{1}{\u03f5}\right)+O\left(1\right)$ϵ approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of faulttolerant quantum circuits.