Virtual Distillation for Quantum Error Mitigation (2011.07064v3)

Abstract: Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring $M$ copies of a noisy state $\rho$. This enables us to estimate expectation values with respect to a state with dramatically reduced error, $\rho^M/ \mathrm{Tr}(\rho^M)$, without explicitly preparing it, hence the name "virtual distillation". As $M$ increases, this state approaches the closest pure state to $\rho$, exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of $\rho$). We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.

Analyzing Virtual Distillation for Quantum Error Mitigation

Quantum error mitigation techniques are indispensable in improving the performance of the current generation of noisy intermediate-scale quantum (NISQ) devices. The paper "Virtual Distillation for Quantum Error Mitigation" introduces a novel approach to error correction known as virtual distillation. This method utilizes multiple copies of a quantum state to estimate expectation values with considerably reduced error, thereby facilitating effective noise suppression.

Virtual distillation circumvents the direct preparation of a purified state, instead enabling the estimation of properties of the closest pure state to the noisy quantum state. This approach leverages entanglement and measurement of multiple copies $M$ of the target state $\rho$. By evaluating the state $\rho^M / Tr(\rho^M)$, errors can be mitigated exponentially quickly as $M$ increases, owing to the state's convergence towards the dominant eigenvector of $\rho$.

Highlights and Numerical Results:

1. Exponential Error Suppression: The effectiveness of virtual distillation is primarily governed by the behavior of the dominant eigenvector of the density matrix. The technique is capable of exponential error suppression across several orders of magnitude, akin to the strategies that use quantum phase estimation and symmetry verification.
2. Numerical Simulations: Through a series of numerical simulations, virtual distillation successfully reduces errors by up to three orders of magnitude. These simulations span various applications, including random quantum circuits and the optimization of Hamiltonian evolutions, such as the Heisenberg spin model. Notably, the degree of error mitigation appears to be enhanced as the system size increases.
3. Stochastic Noise Model: Under a phenomenological model of stochastic noise, this approach demonstrates quadratic error suppression for errors that are orthogonal to the noiseless state. However, it also presents a coherent error floor introduced by shifts in the dominant eigenvector, establishing a limit on achievable correction.
4. Mitigation of Algorithmic Errors: The paper extends the application of virtual distillation beyond hardware-induced errors, exploring algorithmic errors in randomized evolution approaches like qDRIFT. Virtual distillation further optimizes qDRIFT by significantly reducing the coherent steps required to approximate unitary evolution.

Implications and Future Prospects:

The implications of virtual distillation are wide-reaching, encompassing enhancements to quantum computing’s efficacy at both theoretical and practical levels. The approachable nature of the technique ensures that it can be readily applied to existent multi-copy quantum operations with minimal overhead, a promising attribute given the current constraints in quantum hardware.

In terms of future research, the study opens avenues for exploring more complex error models and techniques that incorporate virtual distillation with other quantum error mitigation strategies. Combining distillation with other coherent error-correction methods could offer substantial improvements, particularly as devices transition towards fault-tolerance.

By addressing incoherent errors primarily, virtual distillation complements existing work, positioning itself as a robust tool for those studying quantum dynamics and near-term applications. Enhanced optimization or alternatively structured virtual distillation methods could further improve computational fidelity, edging closer to sustainable quantum advantage in a broader array of settings.

This exploration of virtual distillation marks a significant contribution to advancing quantum error mitigation, reinforcing the foundational understanding of density matrix polynomializing techniques and their practical applicability in NISQ devices.