Exponential distillation of dominant eigenproperties

1. Problem Statement

Estimating the expectation values of observables in specific eigenstates (ground or excited) of quantum many-body systems is a fundamental task in quantum chemistry, material science, and high-energy physics. While quantum computers offer potential exponential advantages, current and near-future devices face significant hurdles:

• Fault-Tolerant Algorithms: Standard algorithms like Quantum Phase Estimation (QPE) require deep circuits and high qubit counts, exceeding the capabilities of early fault-tolerant machines.
• Variational Algorithms: Techniques like VQE are suitable for near-term devices but suffer from issues like barren plateaus, local minima, and high shot-number requirements.
• Initial State Limitations: Many algorithms assume a perfect initial state or are limited to ground states. In practice, initial states often have overlap with multiple eigenstates, and isolating a specific target eigenstate (especially excited states) is difficult.
• Resource Constraints: Existing error mitigation techniques (like Virtual Distillation) often require multiple copies of the quantum state, which is resource-prohibitive for early fault-tolerant devices with limited logical qubits.

The core challenge is to estimate arbitrary observable expectation values in a target eigenstate using a single copy of a quantum state that has only a dominant (but not perfect) overlap with that target, while achieving exponential error suppression without deep circuits.

2. Methodology: Distillation of Dominant Eigenproperties (DDE)

The authors propose a hybrid quantum-classical algorithm called Distillation of Dominant Eigenproperties (DDE). The method combines random time evolution with classical Monte Carlo (MC) integration to virtually purify a mixed state.

Core Workflow:

1. Random Time Evolution:

   • Instead of evolving the state for a fixed time, the algorithm evolves the initial state $|\psi(0)\rangle$ for random times $t$ sampled from a Gaussian distribution $N(0, \sigma)$.
   • Averaging these time-evolved states creates a mixed state $\bar{\rho}$.
   • Key Insight: If the initial state has a dominant overlap with a target eigenstate $|\psi_q\rangle$ and there is a finite energy gap $\Delta$ between the target and other supported eigenstates, the off-diagonal terms in the energy basis of $\bar{\rho}$ decay super-exponentially with $\sigma$. The resulting state is nearly diagonal, with the target eigenstate having the highest probability weight.

2. Virtual Distillation via Classical Post-Processing:

   • Traditional Virtual Distillation (VD) requires $n$ physical copies of the state to estimate $\text{tr}[\rho^n O]$.
   • DDE Innovation: DDE achieves the effect of $n$ copies without requiring $n$ physical registers. It uses a single quantum register to estimate two-time correlators:
     • $A(t, t') = \langle \psi(t) | O | \psi(t') \rangle$
     • $B(t, t') = \langle \psi(t) | \psi(t') \rangle$
   • These correlators are computed on a 2D time grid using Hadamard test circuits (or optimized variants).

3. Monte Carlo Integration:

   • The non-linear functional required for virtual distillation, $\text{tr}[\bar{\rho}^n O]$, is mathematically equivalent to an $n$-dimensional integral over the time variables.
   • The algorithm uses Monte Carlo sampling to evaluate this high-dimensional integral classically.
   • By taking the ratio of the estimated functionals $\frac{\text{tr}[\bar{\rho}^n O]}{\text{tr}[\bar{\rho}^n]}$, the algorithm effectively "purifies" the state, exponentially suppressing contributions from non-dominant eigenstates.

4. Optimization:

   • The authors introduce circuit optimizations that exploit symmetries (e.g., $B(t, t') = B(t', t)^*$) to reduce the number of unique circuit evaluations from $O(N_T^2)$ to $O(N_T)$, where $N_T$ is the number of time grid points.
   • They also propose a variance reduction technique by shifting the observable $O \to O - cI$ to minimize the variance of the MC estimator.

3. Key Contributions

• Single-Copy Virtual Distillation: The primary theoretical breakthrough is demonstrating that virtual distillation can be performed using only one quantum register. The "multiple copies" are simulated virtually through classical post-processing of time-correlators.
• Exponential Error Suppression: The algorithm provides rigorous guarantees that the error in estimating the target eigenstate property scales exponentially with the number of virtual copies ($n$) and super-exponentially with the time evolution window width ($\sigma$).
• General Applicability: Unlike imaginary time evolution methods (which are limited to ground states), DDE uses real-time evolution, making it applicable to any eigenstate (ground or excited), provided the initial state has dominant overlap.
• Resource Efficiency:
  • Circuit Depth: The circuit depth scales as $\tilde{O}(\Delta^{-1})$, comparable to state-of-the-art phase estimation, making it suitable for early fault-tolerant devices.
  • Qubit Count: Requires only a single logical qubit register (plus an ancilla), avoiding the overhead of multi-copy architectures.
• Quantum-Inspired Classical Simulation: The authors demonstrate that the entire DDE framework can be simulated classically using tensor networks (MPS) for systems up to 100 qubits, offering a new "quantum-inspired" classical approach to estimate excited state properties.

4. Results

The paper validates the DDE framework through extensive numerical simulations:

• Exact Simulation (10-qubit Heisenberg Model): Confirmed that statistical fluctuations are suppressed and systematic bias converges to the theoretical bound as $n$ increases. Extrapolation techniques further improved accuracy.
• Early Fault-Tolerant Robustness:
  • Trotter Error: Demonstrated robustness against algorithmic errors from Trotterization. The errors could be mitigated via polynomial extrapolation of the correlators.
  • Gate Noise: Simulations with logical gate errors (depolarizing noise) showed that DDE estimates remain accurate even with noisy circuits, outperforming unoptimized approaches.
• Variational Integration: Combined DDE with VQE. Even when VQE gets stuck in a local minimum (providing an imperfect initial state with ~53% overlap), DDE successfully distilled the ground state properties, exponentially improving the estimate.
• 100-Qubit MPS Simulation: Successfully simulated a 100-qubit random-field Heisenberg chain. The method accurately predicted both ground and excited state energies, converging to results obtained via DMRG and spectroscopy, demonstrating scalability beyond exact diagonalization limits.

5. Significance

• Bridging the Gap: DDE effectively bridges the gap between heuristic variational methods (NISQ era) and fully fault-tolerant algorithms. It leverages the strengths of both: the shallow circuits of variational approaches and the rigorous error suppression of fault-tolerant techniques.
• Practical Quantum Advantage: The algorithm is specifically designed for early fault-tolerant devices. By trading quantum circuit depth for classical post-processing (MC sampling), it aligns with the expected resource constraints of early error-corrected quantum computers (limited logical qubits, but capable of moderate-depth circuits).
• Excited State Access: It provides a practical pathway to study excited states, which are notoriously difficult for tensor network methods and standard QPE without complex state preparation.
• Classical Inspiration: The demonstration of a 100-qubit classical simulation using DDE suggests that this framework can enhance classical computational chemistry and physics tools, even before large-scale quantum computers are available.

In summary, the paper presents a theoretically rigorous and practically viable algorithm that enables the extraction of high-fidelity eigenstate properties from imperfect initial states using minimal quantum resources, marking a significant step toward practical quantum advantage in simulation tasks.