Exponentially tighter bounds on limitations of quantum error mitigation (2210.11505v3)

Abstract: Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing without the heavy resource overheads required by fault tolerant schemes. Recently, error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. Our framework rigorously captures large classes of error mitigation schemes in use today. By relating error mitigation to a statistical inference problem, we show that even at shallow circuit depths comparable to the current experiments, a superpolynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

• The paper proves that error mitigation scales exponentially with qubit number and circuit depth, necessitating super-polynomial sample complexity.
• The paper shows that high entanglement in circuits intensifies noise sensitivity, thereby elevating the demands on mitigation techniques.
• The paper introduces an information-theoretic framework that treats error mitigation as a statistical inference problem, highlighting its fundamental limitations.

Summary of "Exponentially Tighter Bounds on Limitations of Quantum Error Mitigation"

This paper addresses the critical challenges in quantum error mitigation for near-term quantum computing by identifying and analyzing its fundamental limitations. As quantum error correction is currently resource-intensive for implementations in present-day quantum devices, error mitigation emerges as a plausible strategy. Error mitigation minimizes the effects of noise on quantum circuits through classical post-processing, avoiding the need for error correction's extensive overhead. This paper exposes significant constraints inherent in quantum error mitigation and elucidates why existing methods may not scale effectively with system size.

Main Contributions

1. Exponential Sample Complexity:
   • The paper shows that the resource demand for error mitigation scales exponentially with the number of qubits and the circuit depth. Specifically, even at shallow depth, such as poly-logarithmic and logarithmic, the need for quantum observations or samples increases super-polynomially.
   • Two primary results are presented: for local depolarizing noise and non-unital noise models, demonstrating that exponential or super-polynomial sample complexity is needed for effective error mitigation, indicating significant challenges as systems scale.
2. Influence of Entanglement:
   • The paper details how generated entanglement in a quantum circuit influences its susceptibility to noise. Through rigorous theoretical analysis, it is shown that circuits generating extensive entanglement become exponentially sensitive to noise, thereby hindering error mitigation.
   • The authors reveal that for highly entangling circuit ensembles such as unitary 2-designs, the sample complexity rises, highlighting a direct relationship between circuit complexity, entanglement, and error mitigation challenges.
3. Theoretical Framework and Implications:
   • The development of an information-theoretic framework which analyses error mitigation as a statistical inference problem is a novel approach to deriving these fundamental limits.
   • By exploring the theoretical bounds of error mitigation, the paper provides insights into the prospects of quantum advantage in noisy settings, revealing that significant challenges lie ahead.

Implications and Speculations

The research provides a sobering account of the prospects of quantum error mitigation—a tool hoped to advance quantum computing into practical territory without the burdens of full error correction. The exponential limitations elucidated imply that current mitigation schemes may not extend to substantially larger quantum systems without substantial increases in requisite resources. The rapidly spreading entanglement, beneficial for achieving tasks beyond classical capabilities, here, also acts as an antagonist by accelerating noise impact.

Despite these roadblocks, the authors suggest fertile ground for further studies. Future work may explore intermediate schemes blending mitigation and correction, or methods that harness specific circuit localities or structures which might evade these worst-case bounds.

Conclusion

In sum, this paper establishes that while error mitigation currently serves as a promising bridge to full-scale quantum operations, its capabilities are fundamentally curtailed by exponential resource demands in larger, more complex networks. The challenge is clear—without a breakthrough in techniques or theoretical understanding, the path to robust, large-scale quantum computations will continue to heavily rely on formidable error correction strategies.