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Detecting crosstalk errors in quantum information processors

Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, Robin Blume-Kohout

arXiv:1908.09855v3quant-ph

TL;DR

Crosstalk is an emergent multi-qubit failure mode that can violate locality and operation independence, while arbitrary detection is resource-intensive. The paper defines crosstalk operationally and develops an efficient protocol targeting low-weight errors, requiring O(n^3) experiments worst-case and O(n^2) with limited connectivity.

  • Problem

    Crosstalk emerges as quantum processors scale and can violate the spatial locality and independence assumptions underlying well-behaved quantum-information-processing models.

  • Method

    The paper defines crosstalk-free processors through quantum-logic models and operational variables, then detects low-weight crosstalk by testing conditional-independence relations among experimental settings and measurement results.

  • Results

    The protocol was tested on simulated processors with up to six qubits and requires O(n^3) experiments in the worst case or O(n^2) when qubit connectivity is limited.

  • Takeaways & Limitations

    The model-free protocol offers an efficient approach to detecting low-weight crosstalk errors on medium-scale quantum information processors.

  • Takeaways & Limitations

    The efficiency relies on targeting low-weight crosstalk errors, with no guarantee of detecting errors of weight 4 or higher when regions of size at most 2 are used.

Abstract

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Crosstalk occurs in most quantum computing systems with more than one qubit. It can cause a variety of correlated and nonlocal crosstalk errors that can be especially harmful to fault-tolerant quantum error correction, which generally relies on errors being local and relatively predictable. Mitigating crosstalk errors requires understanding, modeling, and detecting them. In this paper, we introduce a comprehensive framework for crosstalk errors and a protocol for detecting and localizing them. We give a rigorous definition of crosstalk errors that captures a wide range of disparate physical phenomena that have been called "crosstalk", and a concrete model for crosstalk-free quantum processors. Errors that violate this model are crosstalk errors. Next, we give an equivalent but purely operational (model-independent) definition of crosstalk errors. Using this definition, we construct a protocol for detecting a large class of crosstalk errors in a multi-qubit processor by finding conditional dependencies between observed experimental probabilities. It is highly efficient, in the sense that the number of unique experiments required scales at most cubically, and very often quadratically, with the number of qubits. We demonstrate the protocol using simulations of 2-qubit and 6-qubit processors.

1 Introduction

As quantum processors scale, crosstalk emerges as a prominent failure mode beyond elementary-operation errors. The paper defines crosstalk errors and develops an efficient, operational detection protocol for many-qubit processors.

  • 1 Introduction: Crosstalk becomes an especially visible failure mode as quantum processors reach roughly 10–20 qubits.Cloud-accessible processors with more than 20 qubits already exist, while systems near 100 qubits may appear within a few years.
  • 1 Introduction: Crosstalk errors are hardware-agnostic deviations that violate spatial locality or independence of quantum operations.The definition focuses on observable quantum-logical effects rather than the underlying physical mechanism.
  • 1 Introduction: The paper defines crosstalk-free quantum processors, constructs a Markovian crosstalk model, and identifies violations as crosstalk errors.It also distinguishes efficiently detectable low-weight errors from arbitrary unknown crosstalk errors that can be difficult to detect.
  • 1 Introduction: The detection protocol uses correlations between experimental settings and outcomes to identify crosstalk structure.Its analysis adapts techniques from causal inference on probabilistic graphical models.
  • 1 Introduction: The protocol requires at most ˜O(n3) experiments and often ˜O(n2) experiments for an n-qubit processor.The paper presents it as a lightweight diagnostic designed for efficient use on many-qubit QIPs.

2 Crosstalk and crosstalk errors

“Crosstalk” encompasses platform-specific physical effects, so the paper introduces “crosstalk errors” to compare their observable quantum-logical consequences across processors. This distinction excludes local errors that could arise from ordinary local noise.

  • 2 Crosstalk and crosstalk errors: Crosstalk traditionally denotes unwanted coupling between signal paths or unintended influence between subsystems of a quantum device.Examples include qubits, control lines, resonators, and photodetectors affecting one another.
  • 2 Crosstalk and crosstalk errors: A single transmon processor can exhibit several distinct crosstalk phenomena, including residual couplings, microwave-line interference, readout coupling, and line noise.These effects are architecture-specific and need not have direct counterparts across platforms.
  • 2 Crosstalk and crosstalk errors: This hardware-agnostic framing enables comparisons between quantum processors without reference to their underlying physical mechanisms.Low-level Hamiltonians and ancillary couplings are not portable across devices.
  • 2 Crosstalk and crosstalk errors: The paper defines crosstalk errors as observable quantum-logical effects that uniquely stem from physical crosstalk.Purely local errors, such as independent bit flips, are excluded because local noise could produce them.

3 Definition of crosstalk errors

The paper defines crosstalk errors through violations of locality or independence in quantum-processor operations. A processor is crosstalk-free when arbitrary circuits satisfy both principles.

  • 3 Definition of crosstalk errors: Crosstalk errors are unwanted dynamics that violate locality, independence, or both.The idealized processor model assumes operations couple qubits or external systems only in a precise and limited scope.
  • 3 Definition of crosstalk errors: Locality requires circuit implementations not to correlate disjoint qubits unless intentional multiqubit operations couple them.This makes the action of an operation on its target qubits well-defined.
  • 3 Definition of crosstalk errors: Independence requires an operation’s evolution on its target qubits to be unaffected by simultaneous operations on disjoint qubits.The condition applies to gates, measurements, and other operations in a circuit.
  • 3 Definition of crosstalk errors: A QIP is crosstalk-free exactly when its behavior on arbitrary circuits satisfies locality and independence.Errors that preserve both principles are not classified as crosstalk errors under this definition.

4 An explicit error model for crosstalk-free processors

The paper defines crosstalk-free Markovian processors through modular layer dynamics: operations must be local and independent of other simultaneous operations. This framework provides a hardware-agnostic basis for classifying, detecting, and quantifying deviations, while recognizing limits from non-Markovianity, gauge freedom, and incomplete categories.

  • Crosstalk-free QIPs: A crosstalk-free Markovian layer factors into tensor products of local component maps that are independent of other gates in the layer.These are the locality and independence conditions defining modular gate operations.
  • Markovian QIPs: Markovian processors model each unique circuit layer as a CPTP map acting on all n qubits.Markovianity makes each layer’s effect well-defined and dependent only on its identity, enabling prediction for new circuits built from characterized layers.
  • Model limitations: Crosstalk detection can confuse non-Markovianity with crosstalk, so non-Markovian effects should be tested before or alongside crosstalk.The Markovian model is expected to remain useful for slightly non-Markovian processors, but detection requires comparing these effects with the protocol’s sensitivity.
  • Hardware-agnostic modeling: The model is hardware-agnostic, replacing system-specific Hamiltonian descriptions with a tractable effective model on n qubits.The CPTP-map formalism supports cross-platform abstraction but is coarse-grained and can sometimes be counterintuitive as a picture of underlying physics.
  • Useful terminology for crosstalk errors: Crosstalk errors include any violations of the crosstalk-free model, including locality violations and independence-only violations.The paper calls these categories absolute and relative crosstalk errors, respectively.
  • Useful terminology for crosstalk errors: The category proposals are not exhaustive, may overlap, and lack specific protocols for rigorously distinguishing all categories.Some violations may fall outside the proposed categories or bridge them.

5 Crosstalk errors are too diverse to detect without assumptions

Arbitrary crosstalk errors can be strong yet computationally difficult to detect, so efficient detection requires assumptions about the error class. The paper focuses on low-weight errors, which admit efficient generic detection and localization while leaving high-weight errors to device-specific protocols.

  • 5.1 Detecting arbitrary crosstalk errors is hard: A relative crosstalk error can affect only one of exponentially many gate layers, making it hard to find despite being easy to demonstrate when that layer is known.The example preserves locality but violates independence because an error on one qubit depends on how all other qubits are controlled.
  • 5.1 Detecting arbitrary crosstalk errors is hard: An absolute crosstalk error can be strong on one input state yet nearly invisible on most states, creating a catch-22 for detection.The phase operation indexed by an unknown n-bit string can be detected with short circuits if the string is known, but locating it requires searching over possible inputs.
  • 5.1 Detecting arbitrary crosstalk errors is hard: Detecting arbitrary strong crosstalk efficiently is impossible because the relevant configurations can grow exponentially with the number of qubits.Characterizing such errors is even harder, so any efficient protocol must restrict the class of errors it targets.
  • 5.2 Low-weight crosstalk errors: The protocol therefore targets low-weight crosstalk errors involving only a few subsystems that should otherwise remain independent.For relative crosstalk, the subsystems include each qubit and its classical control digit, so conditional dependence can appear as a weight-2 error.
  • 5.2 Low-weight crosstalk errors: There are only O(n)k errors of weight at most k on n qubits, supporting efficient detection without exponential resources.Typical error maps may contain higher-weight terms, but their contributions can decline exponentially and often be approximated accurately by low-weight sums.
  • 5.2 Low-weight crosstalk errors: High-weight crosstalk is not expected to be captured generically and is instead associated with architecture-specific control features requiring tailored protocols.Low-weight errors remain plausible across architectures, motivating a generic protocol for detecting and localizing them.

6 An operational protocol for detecting crosstalk errors

The protocol operationalizes crosstalk detection through conditional dependencies among circuit settings and measurement outcomes, targeting low-weight errors with polynomially many experiments. It combines region partitioning, randomized circuit sampling, conditional-independence analysis, and network reconstruction, while noting important scope and statistical caveats.

  • 6.1 Model-free framework and definitions: The model-free framework treats regional circuit settings and measurement outcomes as random variables whose conditional dependencies reveal crosstalk errors.The settings encode preparation, gates, and measurement bases for each region, while outcomes are measured results.
  • 6.1 Model-free framework and definitions: The model-free definition is operationally equivalent to the model-based definition, linking observable random-variable conditions to locality and independence of quantum operations.The model-based account uses CPTP-map conditions, whereas the model-free account uses operational settings and outcomes.
  • 6.2 Defining regions: O(n2 log(n)) randomized partitions detect pairwise crosstalk between any pair of 2-regions with high probability, compared with O(n4) brute-force partitions.Under local connectivity, randomized partitioning improves to O(n log(n)), while brute-force partitioning improves to O(n2).
  • 6.3.1 An explicit construction: The exhaustive circuit design forms a hypercube of all combinations of regional subcircuits, whereas lightweight sampling sparsely fills it using repeated target subcircuits and randomized contexts.A constant Ncon is found sufficient in practice with respect to the number of regions M.
  • 6.5 Discussion and limitations: The protocol requires O(n3 log(n)) distinct experiments in the worst case and O(n2 log(n)) under local connectivity, with tractable post-processing.The protocol’s efficiency targets low-weight crosstalk errors rather than all possible crosstalk errors.
  • 6.3.2 Choosing the subcircuits for each region’s bag: Application-representative subcircuits emphasize crosstalk affecting particular algorithms but may miss errors that arise only when complete application circuits are assembled.Such errors require an application- or architecture-specific test.
  • 6.4 Network discovery: Edges in the learned network indicate crosstalk between regions, not direct causal relationships, because the causal-inference tools are used to detect conditional dependencies.The paper explicitly cautions that the network should not be interpreted as a causal explanation.
  • 6.4.2 Network discovery algorithms: The PC algorithm reconstructs a network by testing conditional independences with increasing conditioning-set sizes, then applying orientation rules to estimate a DAG.Multiple hypothesis testing is not adjusted in the standard PC algorithm, complicating false-positive control.

7 Simulations

Simulations show that the protocol detects and localizes several crosstalk mechanisms in two-qubit processors and identifies expected crosstalk structure in a six-qubit device. Edge weights quantify distributional differences rather than physical error rates, and their interpretation depends on sampled experiments.

  • Two-qubit simulations: The simulations apply the protocol to operation, coherent, and measurement crosstalk models in two-qubit processors.The simulated experiments also include local depolarization errors and low signal-to-noise conditions.
  • Two-qubit simulations: Red edges identify crosstalk by revealing conditional dependencies between variables in different qubit regions.Operation crosstalk produces a red edge from settings in region 0 to results in region 1.
  • Two-qubit simulations: Coherent crosstalk produces conditional dependencies between settings and results across regions and between results in different regions.The simulations report no clear causal direction for this error type.
  • Quantification: Maximum TVD edge weights do not directly measure physical crosstalk because sampled experiments and finite statistics can affect their values.The authors recommend using maximum TVD to identify device regions needing mitigation and to inspect configurations associated with the maximum.
  • Six-qubit simulations: Six-qubit simulations detect crosstalk between vertical neighbors using 300 distinct experiments.The result demonstrates an experimental burden that scales essentially linearly with the number of qubits in this setting.

8 Conclusions

The paper defines crosstalk errors both through hardware-agnostic QIP dynamics and operational variables, then develops a conditional-independence protocol targeting low-weight crosstalk. Simulations up to six qubits support its scalability, while several extensions remain future work.

  • The paper provides a universal, hardware-agnostic definition of crosstalk errors based on representations of gates, preparations, and measurements.
  • It also gives a model-free operational definition and a protocol that detects crosstalk through conditional independence relations among settings and measurement results.
  • The protocol targets low-weight crosstalk errors and requires ˜O(n3) experiments in the worst case and ˜O(n2) in realistic limited-connectivity scenarios.
  • Future work includes evaluating alternatives to the PC algorithm and developing efficient protocols to characterize detected crosstalk errors.

A Conditional versus marginal independence

The paper uses conditional rather than marginal independence to reconstruct cross-region dependencies reliably. Marginal testing can mistake shared causes for direct dependence, especially when settings are not suitably randomized.

  • Conditional independence tests are used to reconstruct the dependency relationships represented in the two-qubit causal graph.
  • Testing marginal independence between R1 and S2 can create a fictitious dependence because both variables share the common cause S1.
  • Marginal independence would suffice if experiment design made S1 and S2 independent, but that randomization may be difficult to guarantee on larger quantum platforms.

B Equivalence of two definitions of crosstalk

The appendix presents two crosstalk-free QIP definitions at different abstraction levels and establishes their equivalence. The operational definition uses conditional independence among experimental settings and outcomes.

  • B Equivalence of two definitions of crosstalk: The first definition characterizes crosstalk-free QIPs through quantum-operation properties, while the second uses conditional independence between classical experimental variables.The two definitions are stated at different abstraction layers: model-based quantum operations versus operationally observed variables.
  • B Equivalence of two definitions of crosstalk: The appendix connects the operational probabilities to quantum states, operations, and measurements using the Born rule, conditioning, and marginalization.The construction uses general two-qubit CPTP maps, POVMs, and an initial state without additional factorization assumptions at this stage.
  • B Equivalence of two definitions of crosstalk: For a two-qubit QIP, the relevant variables are settings S_i and measurement results R_i for each qubit.Settings enumerate single-qubit gate sequences, and outcomes record the corresponding measurement results.
  • B Equivalence of two definitions of crosstalk: The crosstalk-free condition requires P(R_i|S_i, S_j, R_j) = P(R_i|S_i) for distinct qubits i and j.This conditional-independence statement expresses that another qubit’s setting and outcome do not add information about the first qubit’s result.

B.2 Definition 1 ⇒Definition 2

The appendix proves that the model-based crosstalk-free conditions imply the operational conditional-independence conditions. It also frames the reverse direction through a contrapositive argument.

  • B.2 Definition 1 ⇒Definition 2: A model-based crosstalk-free Markovian QIP requires state preparations, gate operations, and measurements to satisfy the specified factorized conditions.These conditions encode the model’s locality and independence assumptions.
  • B.2 Definition 1 ⇒Definition 2: Substituting the factorized operations into the conditional-independence equations shows P(R_i|S_i, S_j, R_j) = P(R_i|S_i).The proof verifies the condition for both qubit orderings.
  • B.2 Definition 1 ⇒Definition 2: Therefore, the model-based definition of a crosstalk-free QIP implies the model-free definition.The implication follows for the operational conditional-independence conditions used in the appendix.
  • B.2 Definition 1 ⇒Definition 2: For the converse, the proof uses the contrapositive that violating locality or independence leads to a violation of the operational definition.The argument explicitly targets violations of the conditional-independence equations.

B.3.1 Locality

The locality proof shows that nonfactorizable state preparations or gate operations violate the operational crosstalk-free condition. The argument also extends to nonlocal measurements by state-preparation and measurement duality.

  • B.3.1 Locality: A Markovian QIP violates locality when its state preparations, gate operations, or measurements fail to factorize.The locality assumption concerns all three operation types in the model.
  • B.3.1 Locality: A nonfactorizable initial state generally produces joint outcome distributions that cannot equal products of marginal distributions.The proof specializes measurements to expose the incompatibility with conditional independence.
  • B.3.1 Locality: Therefore, a nonseparable initial state violates the conditional-independence crosstalk-free condition.This case assumes the other operations factorize.
  • B.3.1 Locality: For nonfactorizable gate operations, the proof assumes factorized state and measurement operators and considers an entangling operation induced by one qubit’s setting.The resulting operation couples the two qubits despite the factorized assumptions elsewhere.
  • B.3.1 Locality: Nonfactorizable gate operations violate the model-free crosstalk-free condition.The proof concludes this from the impossibility of the required equality for all settings and outcomes.
  • B.3.1 Locality: The corresponding result for nonlocal measurements follows from the duality between state preparation and measurement.Thus locality violations across these operation types are linked to operational violations.

B.3.2 Independence

The independence proof considers operations on one qubit that depend on another qubit’s setting. For sufficiently rich settings, such dependence produces an operational violation except in an unlikely invariance case.

  • B.3.2 Independence: Under locality, independence violations occur when the operation on one qubit depends on the other qubit’s setting.The paper represents this dependence using an effective CPTP error map on the first qubit.
  • B.3.2 Independence: The exceptional equality can hold only when the effective error maps act trivially on the measurement effects.The proof explicitly considers nontrivial error maps and identifies this invariance as the alternative condition.
  • B.3.2 Independence: The effective error map captures the action on qubit 1 after factoring out its desired gate sequence.It is induced by a sequence performed on qubit 2.
  • B.3.2 Independence: For sufficiently rich settings, one independence-violating sequence makes additional violating sequences likely, while invariance of all measurement effects is extremely unlikely.This supplies the setting richness used in the proof.
  • B.3.2 Independence: Consequently, independence violations lead to violations of the model-free crosstalk-free definition for a sufficiently rich set of settings.The conclusion applies when the relevant error maps do not act trivially on all measurement effects.

B.4 Definition 1 ⇐⇒Definition 2

The paper proves both directions needed to establish equivalence between its model-based and model-free definitions of crosstalk-free quantum processors.

  • Equivalence between the model-based and model-free definitions of crosstalk-free QIPs is established.
  • The proof establishes both required directions of implication.
  • The result connects the paper’s two definitions of crosstalk-free quantum processors.

C Pseudocode for lightweight experiment design

The lightweight experiment-generation procedure constructs parallel circuit experiments across regions, while the PC algorithm infers conditional-dependence structure by progressively pruning and orienting graph edges.

  • Lightweight experiment design: Algorithm 1 outputs roughly M × Ncircs × Ncon experiments for an M-region QIP.Each experiment contains length-L circuits on every region.
  • Lightweight experiment design: For each region, the procedure samples Ncircs length-L circuits composed of elementary gates.
  • Lightweight experiment design: The design combines sampled circuits across regions and generates Ncon experiments for each selected circuit-region pairing.Other regions may receive idle circuits with probability pidle.
  • Lightweight experiment design: Duplicate experiments are removed from the generated set.
  • PC algorithm: The PC algorithm begins with a complete undirected graph and tests conditional independence over increasingly large subsets of adjacent nodes.It removes an edge when the corresponding variables are conditionally independent given the tested subset.
  • PC algorithm: After pruning, the algorithm forms a skeleton graph and orients its remaining edges using orienting rules.The crosstalk implementation uses an order-independent version of the PC algorithm.
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