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Experimental verification of quantum computations

Stefanie Barz, Joseph F. Fitzsimons, Elham Kashefi, Philip Walther

arXiv:1309.0005v1quant-ph

TL;DR

The paper asks whether a verifier with limited resources can certify a quantum computation whose result it cannot compute itself. It combines blind quantum computing with trap-based verification and demonstrates the protocol using four photonic qubits, obtaining trap correctness above 90% and a Bell-inequality violation exceeding 3 standard deviations.

  • Problem

    The paper addresses whether an entity unable to compute a quantum computer’s results can nevertheless test those results.

  • Method

    The protocol combines interactive proof systems with blind quantum computing and trap computations so a limited-resource verifier can test a more powerful quantum processor.

  • Results

    Trap measurements yielded probabilities well above 90%, and the Bell test violated the classical bound |S| = 2 by more than 3 standard deviations.

  • Takeaways & Limitations

    The work demonstrates experimentally how a verifier can test whether a quantum computer is quantum and whether it computes correctly.

  • Takeaways & Limitations

    The implementation assumes quantum mechanics; a full demonstration without that assumption would require entangled photons sent to two distant laboratories.

Abstract

from arXiv · show

Quantum computers are expected to offer substantial speedups over their classical counterparts and to solve problems that are intractable for classical computers. Beyond such practical significance, the concept of quantum computation opens up new fundamental questions, among them the issue whether or not quantum computations can be certified by entities that are inherently unable to compute the results themselves. Here we present the first experimental verification of quantum computations. We show, in theory and in experiment, how a verifier with minimal quantum resources can test a significantly more powerful quantum computer. The new verification protocol introduced in this work utilizes the framework of blind quantum computing and is independent of the experimental quantum-computation platform used. In our scheme, the verifier is only required to generate single qubits and transmit them to the quantum computer. We experimentally demonstrate this protocol using four photonic qubits and show how the verifier can test the computer's ability to perform measurement-based quantum computations.

INTERACTIVE PROOF SYSTEMS AND BLIND QUANTUM COMPUTING

The protocol combines interactive proof systems with blind quantum computing so a limited-resource verifier can delegate a computation while keeping its content hidden.

  • INTERACTIVE PROOF SYSTEMS AND BLIND QUANTUM COMPUTING: Interactive proof systems replace directly predicting computational results with verifying that a quantum prover can solve the delegated problem.This framework addresses whether restricted-resource entities can test quantum-computational results they cannot compute themselves.
  • INTERACTIVE PROOF SYSTEMS AND BLIND QUANTUM COMPUTING: Blind quantum computing lets a verifier with limited quantum resources delegate computation to a fully capable prover while keeping the data and computation private.The verifier prepares single qubits, sends them to the prover, and receives measurement results for interpretation.
  • INTERACTIVE PROOF SYSTEMS AND BLIND QUANTUM COMPUTING: The verifier hides measurement instructions using random blind phases and outcome-masking bits, preventing the prover from learning the intended rotations.The prover performs measurements according to the supplied basis and returns the results, while the verifier retains the information needed to interpret them.
  • INTERACTIVE PROOF SYSTEMS AND BLIND QUANTUM COMPUTING: Trap qubits can be prepared from blind cluster states and measured in known bases to support later verification of measurement outcomes.The supplied figure describes blind linear and rotated horseshoe cluster states used to prepare trap qubits.

VERIFICATION OF A QUANTUM COMPUTATION

The verification procedure uses hidden trap computations interspersed with target computations to test whether the prover performs measurements correctly and therefore executes the computation correctly.

  • VERIFICATION OF A QUANTUM COMPUTATION: Trap qubits are isolated in known states with predetermined measurement outcomes, so altered measurement results can reveal cheating by the server.The verifier chooses trap measurement angles so the expected outcomes are known in advance.
  • VERIFICATION OF A QUANTUM COMPUTATION: Measurement-based trap preparation verifies correlations among subsets of measurements rather than relying on a single measurement outcome.In the four-qubit demonstration, one error remains undetected, but it cannot change the reported Bell quantity.
  • VERIFICATION OF A QUANTUM COMPUTATION: Randomly interspersing trap and target runs prevents the server from distinguishing verification tests from actual computations and supports verification of the entire computation.Multiple protocol runs are used so the verifier can randomly choose between an actual computation and a verification test.

ENTANGLEMENT VERIFICATION

The protocol extends blind verification to entanglement by hiding both state generation and Bell-measurement settings within a four-qubit zigzag cluster state.

  • ENTANGLEMENT VERIFICATION: The blind zigzag cluster implements the state-generation and Bell-measurement stages within a measurement-based quantum circuit.The figure contrasts the conventional Bell-test sequence with the corresponding blind cluster-state circuit.
  • ENTANGLEMENT VERIFICATION: A blind zigzag cluster state gives the verifier control over whether the input is entangled or separable while concealing that choice from the prover.The choice is controlled through δ4 and θ4, with the edge between qubits 2 and 3 implementing a CPhase gate.
  • ENTANGLEMENT VERIFICATION: The Bell-test measurement settings are hidden in the blind phases and measurement instructions for the cluster-state qubits.The relevant settings are determined by θ1, θ2, θ3 and δ1, δ2, δ3.
  • ENTANGLEMENT VERIFICATION: The prover cannot identify the state configuration or Bell measurements because all relevant choices remain encoded in information hidden by the blind protocol.This concealment is a stated advantage of the probabilistic implementation in which all qubits are measured.

EXPERIMENT

The experiment implements blind cluster-state verification with photonic qubits, tests trap outcomes, and uses a blind Bell test to verify entangling capability under an explicit quantum-mechanics assumption.

  • EXPERIMENT: The photonic setup generates blind cluster states from photon pairs entangled in polarization and mode through spontaneous parametric down-conversion.Photons are used to communicate and process quantum information within the same physical system.
  • EXPERIMENT: The experiment hides Bell-measurement bases using α = π/2, α′ = σz, β = −3π/4, and β′ = −π/4 encoded through blind phases and instructions.The correlation coefficients are calculated from measured coincidence counts before obtaining the S parameter.
  • EXPERIMENT: The Bell-test result violates the classical bound |S| = 2 by more than 3 standard deviations, supporting verification of entangling gates and cluster-state generation.The experiment uses a subset of blind states, with qubits 1 and 4 fixed to |+⟩ and qubits 2 and 3 fully blind.
  • EXPERIMENT: The verification assumes quantum mechanics; without that assumption, a full demonstration would require sending the entangled photons to two distant laboratories.The distant-laboratory arrangement would prevent a classical computer from mimicking the output a priori.

CONCLUSION

The paper develops and demonstrates a general method for verifying quantum computations with applications to current small-scale quantum computers. It positions verification as both a way to certify quantum devices and a tool for fundamental questions in quantum physics and computer science.

  • CONCLUSION: The authors develop a general verification method that can test whether a quantum computer is quantum and whether it computes correctly.They state that the method is supported by both theory and experiment and is readily applicable to current small-scale quantum computers.
  • CONCLUSION: Future large-scale quantum computers and quantum simulators will require verification because their results cannot simply be calculated and checked classically.
  • CONCLUSION: The relationship between verification mechanisms, computational complexity, and the foundations of quantum physics remains an active research topic.The paper notes that the limit of high computational complexity is largely unexplored and that quantum mechanics might break down at some scale of complexity.
  • CONCLUSION: Verification mechanisms may provide a novel toolbox for addressing fundamental questions in quantum physics and computer science.

VERIFICATION OF A QUANTUM COMPUTATION

The verification analysis bounds the probability that a cheating prover introduces an undetected error by using trap qubits. The proof considers the most general possible cheating strategy.

  • VERIFICATION OF A QUANTUM COMPUTATION: Trap qubits are introduced so that errors in a blind quantum computation are either detected or corrected except with bounded probability.The analysis must account for an arbitrary deviation by the prover to establish this guarantee.
  • VERIFICATION OF A QUANTUM COMPUTATION: The proof treats the prover's strategy as an arbitrary deviation, rather than restricting the analysis to a particular attack.

Individual trap qubits

The trap-qubit analysis models the prover's operations and derives an average error-detection bound for a fixed computation. The resulting inequality relates the error probability to the average trap-detection probability.

  • Individual trap qubits: The prover's individual operations can be combined into a single operation B, yielding a circuit representation of the protocol.The measurement outcomes b_i are obtained in the basis {|0⟩, |1⟩}.
  • Individual trap qubits: Defining B′ = BP † rewrites the protocol as the ideal protocol followed by a deviation containing errors introduced by the prover.
  • Individual trap qubits: The verifier's ideal output b is the chosen computation output m bitwise XORed with a verifier-known random bitstring r.
  • Individual trap qubits: For a trap qubit whose expected outcome is r_i, the error probability is evaluated using the fact that m_i = 0 and therefore r_i = b_i.
  • Individual trap qubits: Averaging over trap locations and random outcomes yields an average detection probability used to derive the error bound.
  • Individual trap qubits: ϵ ≤ n⟨t⟩ bounds the probability of an undetected error using the number of qubits and the average trap-detection probability.Here, n is the number of qubits involved in the protocol.

Traps prepared by MBQC

The protocol prepares isolated trap qubits through measurement-based computation and uses their stabilizer correlations to detect deviations that could alter computational outcomes.

  • Trap preparation: Measurement-based computation prepares isolated trap qubits from non-trap qubits, with trap outcomes determined by blind measurement correlations.The possible preparation patterns and corresponding trap states are summarized in Table I.
  • Error analysis: Each trap measurement can be interpreted as a stabilizer measurement, allowing deviations to be classified by whether Pauli terms commute or anticommute with trap settings.Table IV organizes these Pauli terms according to their detectability.
  • Protocol: The protocol randomly selects either a normal computation or a trap computation and randomly chooses a trap index among the available settings.Uniform selection among the three stabilizer measurements is optimal under the stated experimental restrictions.
  • Verification bound: Errors that flip a computational outcome are detected with probability at least p/4, yielding ϵ ≤ 4⟨t⟩/p when ϵ is the probability of flipped outcomes.Here p is the probability of selecting a trap computation and ⟨t⟩ is the probability that a trap computation yields the correct result.
  • Experimental restriction: The verification requires fully blind qubits in principle, but the experiment fixes δ1 and δ4 while retaining a valid verifier choice unknown to the prover.The authentication argument remains valid as long as the prover has no prior information about this restriction.

Experimental settings

The experiment uses the phase and measurement settings specified in Table V to prepare trap qubits across the four-qubit system.

  • Experimental settings: The experiment chooses the blind phases and measurement settings listed in Table V to prepare traps on all qubits.The cited passage identifies the table as the experimental specification for these settings.

ENTANGLEMENT VERIFICATION

The experiment uses blind measurement-based computation to implement a Bell test while preserving verifier-controlled choices of product or entangled input states and measurement settings.

  • Bell statistic: A CHSH Bell quantity is computed as S = |E(α, β) − E(α, β′)| + |E(α′, β) + E(α′, β′)|, with the classical bound S ≤ 2.The correlation coefficients are obtained from coincidence counts for the four outcome pairs in each measurement basis.
  • Blind Bell-test implementation: The blind framework implements a Bell test on a zigzag cluster state, with the cluster generating the state and later measurements realizing the Bell test.The protocol uses blind phases and measurement settings to control both stages.
  • State preparation: The verifier can choose product or entangled input states by changing the combination −δ4 + θ4, including 0 for a product state and π/2 for an entangled state.The demonstration uses the entangled state CPhase|+⟩|−i⟩.
  • Measurement settings: The Bell measurement settings are encoded in blind-phase and measurement-angle differences, with β = −3π/4 and β′ = −π/4 selected through δ3 − θ3.The α and α′ settings are likewise determined by combinations of δ1, δ2, θ1, and θ2.
  • Error tolerance: Simultaneously flipping the first and fourth measurement outcomes leaves the inferred CHSH quantity unchanged because the Bell result depends only on their parity.This establishes invariance to the error form A ⊗C ⊗C ⊗A relevant to the restricted verification scheme.
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