Exponentially tighter bounds on limitations of quantum error mitigation

TL;DR

This work proves fundamental, exponentially-tight limits on quantum error mitigation for near-term devices by recasting mitigation as a state-discrimination problem and analyzing how entanglement-driven scrambling drives noisy outputs toward the maximally mixed state. Using unitary $2$-designs (notably Clifford unitaries) and Pauli-channel noise, the authors bound the quantum relative entropy to the maximally mixed state and show that the required sample complexity grows super-polynomially with the number of qubits, with the onset occurring at depth $D=\tilde{O}(\log\log n)$. They extend the analysis to non-unital noise and argue that ground-state preparation under noise is likewise severely limited, underscoring a tension between entanglement, noise spreading, and error mitigation. The results suggest that scalable near-term quantum computation will need intermediate error-correction strategies or circuit-design principles that constrain entanglement growth to remain tractable for mitigation.

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.

Figures (3)

• Figure 1: Intuition for our circuit construction: the higher the weight of a Pauli string, the more sensitive it is to Pauli noise, as showcased in Eq. \ref{['eq:noisesensitivity']} in the Supplementary Material. Whereas for product pure states there are Pauli strings with constant-order expectation values for all weights, most quantum states only have correlators of high weight. Thus, most random states are significantly more sensitive to noise than product quantum states. (a) The effect of applying a noiseless Pauli mixing circuit to a computational basis state is to shift the binomial of contributions to a weighted binomial (compare Eq. \ref{['eq:purity_compstate']} to Eq. \ref{['eq:Panel1b']}). (b) The effect of applying one subsequent layer of depolarizing noise on the output of the aforementioned circuit (compare Eq. \ref{['eq:Panel1b']} to Eq. \ref{['eq:Panel2']}). (c) The effect of applying yet another Pauli mixing layer to the state output by the aforementioned circuit (as captured in Eq. \ref{['eq:Panel1b']}).
• Figure 2: To lower-bound the sample complexity of weak error mitigation, we show that it can be used as a subroutine to solve a constructed problem of distinguishing states under noise.
• Figure 3: (a) Idealization of near-term quantum algorithms without quantum noise. Most such algorithms work by running an $n$-qubit quantum circuit $\mathcal{C}$ on an input quantum state $\rho$, measuring the output state, and then returning either samples from the resulting probability distribution or expectation values of specified observables. (b) The model of error mitigation used in this work, building on the framework established in Ref. TEMG21. The quantum channel $\mathcal{C}'_i$ represents the $i^{\text{th}}$ run of $\mathcal{C}'$, the noisy version of $\mathcal{C}$. We model the noise acting on $\mathcal{C}$ by interleaving its layers with layers of a given noise channel. In Section I of the Supplemental Material, we show that this model applies to practical error mitigation protocols such as virtual distillationhuggins2021virtualkoczor2021exponential, Clifford data regression (CDR) czarnik2021error, zero-noise extrapolation (ZNE) PhysRevLett.119.180509 and probabilistic error cancellation (PEC) PhysRevLett.119.180509endo2018practical. In this work, we study how $m$, the number of noisy circuit runs, scales with $n$ and $D$ to reliably recover the expectation values.

Theorems & Definitions (22)

• Theorem 1: Number of samples for mitigating depolarizing noise scales exponentially in the number of qubits and depth
• Theorem 2: Number of samples for mitigating non-unital noise scales exponentially in the number of qubits and depth
• Theorem 3: Resource cost of successful error mitigation
• Theorem 4: Exponentially-many observables (in the same eigenbasis) are needed to output samples from the same basis
• Lemma 1: Purity controls relative entropy to the maximally mixed state
• proof
• Definition 1: Pauli group and Pauli weight
• Definition 2: Depolarizing channels
• Lemma 2: Action of depolarizing noise on a Pauli string depends on its weight
• proof