Exponential distillation of dominant eigenproperties

TL;DR

This work introduces the distillation of dominant eigenproperties (DDE), a hybrid quantum-classical algorithm that estimates observable expectation values in a target eigenstate using a single quantum register and random time evolution to form a nearly diagonal mixed state. By applying virtual distillation on time-averaged states and performing high-dimensional MC integration over two-time correlators, DDE achieves exponential error suppression tied to the spectral gap \\Delta and Gaussian window \\sigma$, with circuit-depth scaling comparable to phase estimation. The authors provide rigorous bounds, demonstrate robustness to Trotter and gate-noise errors, and validate the approach across exact simulations, near-term quantum implementations, variational simulations, and quantum-inspired classical tensor-network simulations up to 100 qubits. They also show that DDE can extend to excited-state properties beyond ground-state energy, offering a potentially practical pathway toward quantum advantage in quantum chemistry, materials science, and beyond. The framework is flexible with respect to initial-state preparation and time-evolution methods and integrates well with tensor-network techniques for classical simulations in regimes where those methods remain efficient.

Abstract

Estimating observable expectation values in eigenstates of quantum systems has a broad range of applications and is an area where early fault-tolerant quantum computers may provide practical quantum advantage. We develop a hybrid quantum-classical algorithm that enables the estimation of an arbitrary observable expectation value in an eigenstate, given an initial state is supplied that has dominant overlap with the targeted eigenstate -- but may overlap with any other eigenstates. Our approach builds on, and is conceptually similar to purification-based error mitigation techniques; however, it achieves exponential suppression of algorithmic errors using only a single copy of the quantum state. The key innovation is that random time evolution is applied in the quantum computer to create an average mixed quantum state, which is then virtually purified with exponential efficacy. We prove rigorous performance guarantees and conclude that the complexity of our approach depends directly on the energy gap in the problem Hamiltonian and remarkably, can be compared to phase estimation combined with amplitude estimation in terms of its scaling with respect to a target precision. We demonstrate in a broad range of numerical simulations the applicability of our framework in near-term and early fault-tolerant settings. Furthermore, we demonstrate in a 100-qubit example that direct classical simulation of our approach enables the prediction of ground and excited state properties of quantum systems using tensor network techniques, which we recognize as a quantum-inspired classical approach.

Figures (15)

• Figure 1: DDE proceeds by estimating two-time correlators $A(t,t') = \bra{\psi(t)} O \ket{\psi(t')}$ and $B(t,t') = \bra{\psi(t)} \ket{\psi(t')}$ using Hadamard-test circuits over a 2D time grid in $t$ and $t'$. Then, Monte Carlo sampling is used to evaluate high-dimensional integrals, which lead to the estimation of an observable expectation value in a target eigenstate of a problem Hamiltonian. \ref{['theo:joint-bound']} guarantees that the error $\mathcal{Q}$ of the final estimate is suppressed exponentially with respect to chosen hyperparameters.
• Figure 2: Numerically verifying our error bounds. Using an initial state with $p_q \approx 0.85$ and a time grid defined on $[-5\sigma, 5\sigma]$ with step size $dt = 1$, we apply temporal averaging in a $12$-qubit random-field Heisenberg model simulation. Error contributions are shown against the standard deviation $\sigma$. (left) Verifying \ref{['lemma:time-evolution']}: numerically computed error $\mathcal{E} =\bar{\rho}-\rho$ norms (dashed lines) are indeed below their corresponding analytical upper bounds (solid lines), and decrease super-exponentially as we increase the standard deviation of the normal distribution. (right) Numerically demonstrating \ref{['theo:joint-bound']}, the error scaling, as measured in the trace norm of $\eta = \bar{\rho} ^n / \mathrm{tr}[\bar{\rho} ^n] - \ketbra{\psi_q}$, is shown for an increasing number of copies, where $\ket{\psi_q}$ denotes the target eigenstate. As stated in \ref{['theo:joint-bound']}, this error norm is a sum of two terms: the first term is suppressed exponentially as we increase the number of copies $n$, whereas the second term is suppressed super-exponentially as we increase $\sigma$, as illustrated above.
• Figure 3: Quantum circuit that enables estimating eigenstate properties, directly following \ref{['lemma:time-evolution']} and \ref{['lemma:esd']}. The initial state $\ket{\psi(0)}$ is time-evolved for a randomly sampled duration in each of the registers before a derangement (denoted $D_n$) and the observable $O$ are applied, both conditioned on the state of the ancilla qubit. While this implementation requires $n$ quantum registers, DDE only requires $1$ register for any $n$ by using either the circuits shown in \ref{['fig:algo']} or \ref{['fig: QC']}.
• Figure 4: Verifying DDE in a $10$-qubit random-field Heisenberg exact simulation. (left) Mean errors $\Delta\langle O \rangle$ (estimated from $10^4$ independent runs) for an increasing number $n$ of virtual copies (solid lines) converge for an increasing number of MC samples to a systematic bias consistent with \ref{['theo:joint-bound']}. (right) Plotting the systematic bias for an increasing number $n$ of virtual copies and extrapolating to $n\rightarrow \infty$ via an exponential model function improves our estimate.
• Figure 5: Trotter simulation compared with exact time evolution in an $8$-qubit Fermi-Hubbard model. (left) Mean errors in the estimated expectation value (using $10^6$ MC samples and averaging over $10^3$ runs) for an increasing number $M$ of Trotter steps and for an increasing number $n$ of copies. Mean errors using Trotterization (solid lines) rapidly (in $M$) converge to errors using exact time evolution (dashed lines) confirming the robustness of DDE to algorithmic errors. (right, inset) Applying polynomial extrapolation to the real (solid line, circles) and imaginary (dashed line, squares) parts of the single matrix entries $A(169.5, 93.5)$ (blue) and $B(-31.5, -70.5)$ (orange). The ideal values are also shown for comparison (black triangles). (right) Mean errors in the expected value estimation using correlators $A$ and $B$ estimated using extrapolated entries (solid lines) closely approximate mean errors obtained via exact time evolution (dotted lines) confirming the efficacy of extrapolation as an algorithmic error mitigation technique.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2: Theorem 2 of Ref. koczor2021exponential
• Theorem 1
• Lemma 3
• Proposition 1
• Lemma 4
• Theorem 2
• proof
• proof
• proof