Error mitigation can improve estimates from noisy logical quantum circuits, but it cannot generally make arbitrarily deep universal computation cheap. For several defined noise models and classes of mitigation protocols, theory proves that the samples needed for a target accuracy can grow exponentially with circuit depth; some analyses also find width-dependent costs. These are conditional lower bounds, not one universal run-count forecast. Experiments show mitigation can help on tested logical circuits, but it does not replace quantum error correction.
What limits error mitigation in a universal logical circuit?
The central limit is sampling cost. Error mitigation uses results from noisy executions to estimate what an ideal circuit would have produced. Because the executions are noisy, the estimate can require many repetitions. In important mathematical settings, the number of samples needed to reach a specified accuracy grows exponentially with circuit depth. A separate line of work shows that circuit width can also worsen the cost under particular assumptions.
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That finding is serious, but it is not a theorem that every universal circuit, every hardware platform, or every mitigation method has the same exponential cost. Each result applies to a stated noise model, circuit class, estimator objective and set of allowed protocols. The literature establishes fundamental obstacles for broad, specified classes—not a single numerical limit for all logical quantum computation.
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Repair Windows errors before they cause bigger problemsFix Now →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Clear out junk files and repair common Windows errorsFree Scan →“Universal” describes a circuit model capable of expressing general quantum computations to a desired approximation; it does not by itself specify circuit depth, width, error rates or noise correlations. “Logical” means information is encoded across physical qubits so that operations and measurements represent encoded qubits. Logical encoding can reduce errors, but it does not make them vanish automatically. The relevant question is therefore not simply whether a circuit is universal or logical, but what noise remains and what accuracy the user needs from the final estimate.
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How mitigation works—and why sampling becomes costly
A quantum circuit produces measurement outcomes, not a complete classical description of its quantum state. A user estimates quantities of interest—such as expectation values—from repeated executions. Noise changes the outcome distribution and can bias those estimates away from the ideal circuit’s result.
Mitigation attempts to infer the ideal quantity from noisy data, often by combining results from executions under altered conditions or by statistically reweighting noisy outcomes. This can reduce bias, but the estimate still has uncertainty. To make that uncertainty smaller, more samples are generally needed. Some approaches also amplify statistical fluctuations: correcting a larger predicted noise effect can make each effective sample carry less useful information. In that case, accuracy becomes increasingly expensive in runs.
Two commonly discussed ideas help illustrate the tradeoff, though they do not cover every protocol addressed by the lower-bound papers:
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- Probabilistic error cancellation (PEC) uses a noise description to construct a statistical combination of operations whose average represents the intended, less-noisy operation. The correction is probabilistic, so its variance—and hence its sampling requirement—can grow as the circuit gets longer. Calibration or an accurate noise model is consequential: if the model is wrong, the corrected estimate can remain biased.
- Zero-noise extrapolation (ZNE) collects estimates at different effective noise levels and extrapolates toward a zero-noise value. The result depends on the noise-scaling and extrapolation procedure. Extrapolation may lower observed error for a tested workload, but that does not establish that the method will retain affordable variance at unbounded depth.
Mitigation is thus a trade: less bias in the estimate may require more samples, stronger assumptions about noise, or both. “More accurate” is not enough to establish scalability; the cost to obtain that accuracy matters.
What the depth lower bounds establish
The 2023 Physical Review Letters paper “Universal Sampling Lower Bounds for Quantum Error Mitigation” derives lower bounds on sampling cost for a wide class of protocols, including protocols with nonlinear postprocessing. Its abstract says the cost required to mitigate errors grows exponentially with circuit depth for various noise models. The authors describe the result as a universal lower bound on the sampling cost needed to achieve desired accuracy with high probability.
The words “wide class” and “various noise models” matter. The result identifies a fundamental scalability obstacle across the specified settings; it does not say that all protocols under all conceivable noise conditions share one identical cost function. Nor does it give a universal number of circuit runs that applies independently of the desired accuracy or the task.
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A related 2022 npj Quantum Information analysis, “Fundamental limits of quantum error mitigation”, studies layered circuits under local depolarizing noise and finds exponential overhead with depth for its general protocol class. It also establishes optimality of PEC for a specified local-dephasing setting and protocol family. These results are not interchangeable: depolarizing and dephasing noise are different models, and the optimality claim is limited to its stated setting. Together, the papers show why a depth claim must be read alongside its assumptions about noise and admissible mitigation protocols.
Exponential growth is an asymptotic statement: as the relevant circuit depth increases, the lower bound grows exponentially under the result’s conditions. It is not a claim that every short circuit is already impractical, and the cited work does not provide a single measured sample count for every circuit. Whether the bound rules out a particular application depends on the depth range, target precision, hardware noise and available sampling budget.
Why width can matter as well as depth
Depth counts sequential circuit layers or operations along the computation; width describes how many qubits participate. A circuit can be wide but comparatively shallow, or narrow and deep. Treating depth as the only relevant scale misses an additional limit identified in later work.
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The 2024 Nature Physics paper “Exponentially tighter bounds on limitations of quantum error mitigation” develops stronger bounds that include circuit-width dependence in specified settings. The analyzed cases include local depolarizing noise and highly entangling circuits, and the article also considers non-unital noise. The assumptions are part of the result: it would be inaccurate to turn this into “all wide circuits always require exponential overhead.”
For practical evaluation, width and depth should be considered together with the circuit’s structure. A width-dependent bound for highly entangling circuits under a given model does not automatically apply to a sparse or weakly entangling workload, and a depth lower bound derived for one noise structure cannot simply be reassigned to another. The sources establish that these dimensions can matter; they do not supply a single formula that predicts the cost of every workload.
What changes when the qubits are logical?
Logical qubits and error mitigation address errors in different ways. Quantum error correction (QEC) encodes quantum information in multiple physical qubits and uses syndrome information and recovery procedures to suppress errors during computation. Error mitigation instead estimates idealized results from noisy execution data. In a logical circuit, mitigation may target residual logical errors after encoding and correction; using it does not mean the circuit is free of noise.
Theoretical work has examined combining QEC with mitigation rather than treating them as alternatives. The 2022 PRX Quantum paper “Quantum Error Mitigation as a Universal Error Reduction Technique” proposes a framework combining QEC with PEC in an early-fault-tolerant setting. In two regimes considered by the authors, their model estimates reductions of 80% and 45% in required physical qubits relative to the comparison used in that study.
Those percentages are modeled, regime-specific resource estimates, not experimentally observed savings that apply to all devices or workloads. They show that a carefully chosen combination may improve a resource tradeoff in the modeled settings; they do not establish that mitigation can take the place of QEC for arbitrarily long universal computation.
There is also experimental evidence at the logical-circuit level. A 2025 Nature Communications study, “Demonstrating quantum error mitigation on logical qubits”, applies ZNE to quantum-error-correction circuits and reports reduced logical errors across the tested repetition-code and surface-code circuits and implementation. This is an experimental demonstration on those circuits—not a proof of unlimited scaling, and not a result that overrides the theoretical sampling lower bounds.
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How to judge a claim that mitigation scales
A reported error reduction is meaningful only when read with its cost and scope. When assessing a paper, benchmark or proposed logical-computing architecture, check the following:
- What quantity is being estimated? An expectation value or other target estimator has a precision requirement; “improved result” without an accuracy target does not determine the sampling cost.
- How do depth and width enter? Record whether the result concerns depth, width, both, or a restricted circuit family such as highly entangling circuits.
- Which noise is assumed? Local depolarizing, local dephasing and non-unital noise are distinct assumptions. Noise correlations and whether the model reflects the implementation also affect how a result should be interpreted.
- What protocols are covered? A lower bound for a broad but defined protocol class is stronger than a claim about one method, but it is not automatically a statement about protocols outside that class.
- What does the evidence represent? Separate an asymptotic theorem, a resource model and an experiment. They answer different questions: what cost is unavoidable under assumptions, what a proposed architecture may save in selected regimes, and what worked on a particular implementation.
- What happens to variance and bias? Check whether mitigation reduces bias at the cost of larger variance, and what noise-calibration or model assumptions are needed for the estimate.
- Is the target physical or logical error? A result on physical qubits does not automatically predict the logical-qubit performance of an encoded circuit, or vice versa.
This checklist prevents a common mismatch: comparing a theorem’s worst-case or asymptotic statement directly with one successful finite-size experiment. A finite circuit may benefit from mitigation even when a lower bound warns that the approach cannot remain economical as depth grows. Conversely, observing lower logical error on selected circuits does not show that a fixed sampling budget will suffice as circuits become wider and deeper.
Does error mitigation replace fault tolerance?
No. The literature described here supports mitigation as a potential complement to QEC, including in logical and early-fault-tolerant settings, but does not show that it can generically make arbitrarily deep universal circuits affordable without fault-tolerant protection. Sampling lower bounds are a reason to take cost seriously; they are not a claim that mitigation is useless on every circuit. Experiments and resource models show possible benefits within bounded, specified settings, not universal scalability.
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