Real-Time Evolution for Ultracompact Hamiltonian Eigenstates on Quantum Hardware

In this work we present a detailed analysis of variational quantum phase estimation (VQPE), a method based on real-time evolution for ground- and excited-state estimation on near-term hardware. We derive the theoretical ground on which the approach stands, and demonstrate that it provides one of the most compact variational expansions to date for solving strongly correlated Hamiltonians, when starting from an appropriate reference state. At the center of VQPE lies a set of equations, with a simple geometrical interpretation, which provides conditions for the time evolution grid in order to decouple eigenstates out of the set of time-evolved expansion states, and connects the method to the classical filter-diagonalization algorithm. Furthermore, we introduce what we call the unitary formulation of VQPE, in which the number of matrix elements that need to be measured scales linearly with the number of expansion states, and we provide an analysis of the effects of noise that substantially improves previous considerations. The unitary formulation allows for a direct comparison to iterative phase estimation. Our results mark VQPE as both a natural and highly efficient quantum algorithm for ground- and excited-state calculations of general many-body systems. We demonstrate a hardware implementation of VQPE for the transverse field Ising model. Furthermore, we illustrate its power on a paradigmatic example of strong correlation (${\mathrm{Cr}}_2$ in the def2-SVP basis set), and show that it is possible to reach chemical accuracy with as few as approximately $50$ time steps.

Popular Summary

One of the most promising expected applications of near-term quantum computers lies in the study of static and dynamical properties of quantum many-body systems. Many quantum computing algorithms have been proposed with this goal in mind, with a focus on Hamiltonian eigenvalue extraction, a problem central to chemistry, physics, and materials science. However, the majority of established quantum algorithms require a prohibitively large number of resources for near-term hardware.

Here we show that quantum algorithms relying on short real-time evolution for energy eigenvalue determination, which we refer to as variational quantum phase estimation (VQPE), represent a promising solution for this challenge. Real-time evolution is native to quantum hardware, making these algorithms particularly suited for the near term. Additionally, using a novel transformation, we prove that we can drastically reduce the number of measurements needed from quadratic to linear, marking this a significant improvement over previous quantum algorithms based on time evolution. We show that remarkably few time evolution steps, computed on quantum hardware, are needed to converge both ground- and excited-state energies to experimentally meaningful accuracies. We demonstrate the power of this approach classically on a paradigmatic example of strong correlation, the Cr2 dimer, as well as on quantum hardware for the transverse field Ising model.

Given the capability of VQPE to efficiently extract Hamiltonian eigenstates, we anticipate that this algorithm will become one of the standard approaches for energy estimation on near-term quantum hardware.

Figure 1

Relative error of the first four eigenvalues from the real-time NOVQE secular equation for a Hamiltonian of linear spectrum $E_N=N\mathrm{\Delta }E$ ($\mathrm{\Delta }E=0.75$ here) and different time steps $\mathrm{\Delta }t$ as a function of the number of time steps. The reference state follows $|{\mathrm{\Psi }}_0⟩\propto \sum _N{e}^{-E_N}|N⟩$, such that only the 16 Hamiltonian eigenstates of lowest energy are part of the support space, choosing a coefficient threshold of ${10}^{-12}$. When solving the secular equation, we choose the same threshold for the SVD decomposition of the overlap matrix. The vertical dashed line marks the 15th time step, after which we have as many expansion states as vectors in the support space. The large subplot corresponds to the smallest time step that fulfills the phase cancelation condition, Eq. (10), which reduces to a single condition for this Hamiltonian. The insets in each subfigure correspond to a geometric representation of the phase cancelation condition, with each phase $e^{-i{t}_j\mathrm{\Delta }E}$ a point in the unit circle on the complex plane. See the text for details.

Figure 2

Relative error and corresponding singular values of the overlap matrix for a Hamiltonian of linear spectrum with different noise values. Upper panels: relative error of the first four eigenvalues from the VQPE secular equation for a Hamiltonian of linear spectrum $E_N=N\mathrm{\Delta }E$ ($\mathrm{\Delta }E=0.75$ here) and perfect time step $\mathrm{\Delta }{t}_P$, including Gaussian noise $\mathcal{N}(0,ϵ)$ to the Hamiltonian and overlap matrix elements. We choose the singular value truncation threshold $s_{\mathrm{SV}}$ to be at least 2 orders of magnitude larger than the noise standard deviation $ϵ$. The reference state follows $|{\mathrm{\Psi }}_0⟩\propto \sum _N{e}^{-E_N}|N⟩$, the effective support space size determined using the singular value truncation threshold $N_{\mathrm{SVD}}^{\max }=-\ln (s_{\mathrm{SV}})/(2\mathrm{\Delta }E)-1$. Lower panels: corresponding singular values of the overlap matrix as a function of time. The horizontal dashed line represents the threshold $s_{\mathrm{SV}}$. Note that, until a given singular value is larger than $s_{\mathrm{SV}}$, we cannot extract the corresponding eigenvalue from the generalized eigenvalue problem. This is represented by the horizontal lines in the upper panels. See the text for details.

Figure 3

Relative error and corresponding singular values of the overlap matrix for a Hamiltonian of linear spectrum for a large enough number of time steps to extract eigenstates below the error threshold. Upper panel: relative error of the first five eigenstates from the VQPE secular equation for a Hamiltonian of linear spectrum $E_N=N\mathrm{\Delta }E$ ($\mathrm{\Delta }E=0.75$ here) and perfect time step $\mathrm{\Delta }{t}_P$, including Gaussian noise $\mathcal{N}(0,{10}^{-2})$ to the Hamiltonian and overlap matrix elements. We choose the singular value truncation threshold $s_{\mathrm{SV}}=9\times {10}^{-1}$. The reference state follows $|{\mathrm{\Psi }}_0⟩\propto \sum _N{e}^{-E_N}|N⟩$. Lower panel: corresponding singular values of the overlap matrix as a function of time. The horizontal dashed line represents the singular value threshold $s_{\mathrm{SV}}$. The time step at which each singular value becomes larger than $s_{\mathrm{SV}}$ is marked with a vertical dashed line, connecting upper and lower panels. See the text for details.

Figure 4

Four-qubit version of the asymptotically optimal phase estimation circuit.

Figure 5

Accuracy comparison between VQPE and the Heisenberg limit for the ground-state energy of LiH. Varying numbers of samples per expectation value are shown for VQPE alongside the idealization of VQPE without finite-sampling errors. We use SVD cutoffs of 2, 0.9, and 0.3 respectively corresponding to ${10}^3$, ${10}^4$, and ${10}^5$ samples and ${10}^{-6}$ for the ideal case. All methods use $\mathrm{\Delta }t=0.5$. The top and bottom plots show the maximal evolution time (i.e., how many time steps need to be included for energy convergence), related to the circuit depth, and the total evolution time, i.e., the sum of all time segments needed.

Figure 6

Accuracy comparison between VQPE and QPE for the ground-state energy of the ten-qubit transverse field Ising model with first-order Trotterization and $\mathrm{\Delta }t=0.05$. We use SVD cutoffs of 1, 0.6, and 0.2 respectively corresponding to ${10}^3$, ${10}^4$, and ${10}^5$ samples and ${10}^{-5}$ for the ideal case. Dotted lines show QPE and idealized VQPE results without Trotter errors. The reference state, $|\mathrm{\Phi }⟩=|\varphi {⟩}^{\otimes 10}$, is a product state the maximizes overlap with the exact ground state, $|⟨{\mathrm{\Psi }}_0|\mathrm{\Phi }⟩{|}^2\approx 0.014$ for $|\varphi ⟩\approx 0.979|0⟩+0.205|1⟩$.

Figure 7

A schematic picture of recent eigenvalue algorithms relying on time evolution proposed for quantum computers. The plane corresponds to the complex time plane, with the horizontal axis being real time and the vertical axis imaginary time. The colored dots denote expansion states $|{\varphi }_{j,I}⟩$. The operator $f(H;j)$ denotes any unitary that commutes with the Hamiltonian, and thus the expansion states formed by applying this unitary will remain in the support space and can be used to extract eigenstates. The real-time axis is where the phase cancelation picture applies. On the imaginary-time axis, the coefficients of the excited states are suppressed as time increases, leading to the ground state. The dotted lines indicate that short time imaginary- and real-time states will be very similar, as argued in the short time Krylov method [51]. We emphasize that combinations of states generated on the entire real- or imaginary-time plane have essentially the same support space and can be used as a basis to solve the generalized eigenvalue problem.

Figure 8

Hadamard test circuit, which computes the real and imaginary parts of the overlap matrix, where $|\mathrm{\Omega }⟩$ denotes the system and the top qubit is the ancilla. Here $U=e^{-itH}$.

Figure 9

Ground- (solid) and excited-state (dashed) energies of LiH as a function of the time step (left) and total simulation time (middle). The initial reference vector is the Hartree-Fock state. We use the 3-21g basis set that has 4 electrons and 11 orbitals. The SVD cutoff is ${10}^{-1}$ for both the left and middle plots and the legend is the same for both. The right figure shows the total time to reach chemical accuracy as a function of SVD.

Figure 10

(a) Number of variational parameters to reach chemical accuracy using ASCI (a classical selected configuration interaction approach), and the VQPE algorithm with different $s_{\mathrm{SV}}$ values. These parameters correspond to determinants in ASCI, and to expansion states in VQPE. (b) Comparison of convergence for all different Hilbert space sizes of ${\mathrm{Cr}}_2$. (c) Convergence for all the molecules as a function of total time ($s_{\mathrm{SV}}={10}^{-1}$) with the inset showing convergence of the first excited state that has nonzero overlap with the reference state ($s_{\mathrm{SV}}={10}^{-6}$).

Figure 11

The top figure shows the ground-state energy as a function of time step for the ten-qubit transverse field Ising model run on IBM’s qasm simulator with $s_{\mathrm{SV}}={10}^{-1}$. The middle figure shows results for hardware runs (IBM’s Paris machine) for a two-qubit transverse field Ising model. For both the top and middle plots, VQPE and the Trotter step size are the same, $\mathrm{\Delta }t=0.05$. Classical processing in the middle plot (for both QPU and simulator data) used SVD cutoff $s_{\mathrm{SV}}=2$. The bottom figure shows results for hardware runs (IBM’s Montreal machine) for a three-qubit transverse field Ising model, VQPE $\mathrm{\Delta }t=0.1$, Trotter step size $\mathrm{\Delta }t=0.05$, and $s_{\mathrm{SV}}=1$.