Source-linked AI summary
Topological quantum computing with a very noisy network and local error rates approaching one percent
Naomi H. Nickerson, Ying Li, Simon C. Benjamin
TL;DR
The paper addresses whether scalable networked quantum computers can tolerate very noisy links without requiring prohibitively accurate cells. It introduces a topological noisy-network protocol based on shared GHZ resource states and finds local-error thresholds above 0.82% with 10% network errors. The study does not explicitly simulate multi-logical-qubit computation and assumes equally likely error types.
Problem
Prior noisy-network approaches required cell error rates of about 0.1%, potentially making purification impractical despite networks offering a scalable architecture.
Method
Cells generate shared GHZ states from purified Bell pairs and consume them to perform stabilizer measurements for a 2D toric-code encoding.
Results
At a 10% network error rate, local-error thresholds are about 0.6% for EXPEDIENT, 0.775% for STRINGENT, and 0.825% for STRINGENT+.
Takeaways & Limitations
The protocol largely closes the gap between error tolerance in noisy-network and monolithic architectures, supporting networked approaches based on cells such as ion traps and NV centres.
Takeaways & Limitations
The study does not explicitly simulate computation involving two or more logical qubits and assumes all forms of error are equally likely.
Abstract
from arXiv · showhide
A scalable quantum computer could be built by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. A solution is for cells to repeatedly communicate with each other and so 'purify' any imperfections; however prior studies suggest that the cells themselves must then have prohibitively low internal error rates. Here we describe a method by which even error-prone cells can perform purification: groups of cells generate shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (>=10% error rate) we find that our protocol can succeed provided that intra-cell error rates for initialisation, state manipulation and measurement are below 0.82%. This level of fidelity is already achievable in several laboratory systems.
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2773
The paper develops a noisy-network architecture in which cells use purified shared resource states to measure stabilizers of a topologically encoded quantum memory. With 10% network error rates, simulations find that several protocol variants tolerate local errors approaching one percent.
- Topological codes encode logical qubits collectively and use repeated stabilizer measurements to detect errors and perform logical operations.
- Prior noisy-network analyses required cell error rates of approximately 0.1%, motivating a less demanding purification approach.
- The protocol generates a four-cell GHZ state from purified Bell pairs, then consumes it in one step to measure a four-qubit stabilizer.
- At 10% network error, local-error thresholds are about 0.6% for EXPEDIENT, 0.775% for STRINGENT, and 0.825% for STRINGENT+.
- The model assumes a 10% network error probability and uses controlled-Z, controlled-X, and single-qubit X-basis measurement as intra-cell operations.
- Thresholds are determined by simulating toric-code stabilizer cycles across network sizes, decoding the noisy outcomes with Edmonds’ minimum weight perfect matching algorithm.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2773 ARTICLE
The protocol models imperfect stabilizer measurements as mixtures of correct and incorrect parity projections combined with Pauli errors, then uses numerical simulation and decoding to determine thresholds.
- Error bias toward selected Pauli components may be exploitable by adapting the protocols, potentially increasing error thresholds.The protocol deliberately minimizes σx and σy errors relative to σz because the toric code handles the resulting error type relatively easily.
- Ideal even Z-stabilization projects onto states with definite even parity in the Z basis, while X stabilizers use definite parity in the X basis.The even Z projector includes states such as |0000⟩ and |0011⟩.
- The real projector mixes correct and incorrect stabilizer projections with possible Pauli errors, whose relative weights depend on the underlying error rates.The weights are derived using the Choi-Jamiolkowski isomorphism, including a twirling operation.
- For an even reported outcome, the largest term corresponds to the correct even projector without data-qubit errors.The next-largest term is the pure wrong projection, pairing the same no-error operator with the odd projector.
- The superoperator for the complete stabilizer protocol is simulated with correlated random errors, and Edmonds’ minimum weight perfect matching algorithm resolves the resulting stabilizer flips.The same model gives a monolithic-architecture threshold near 0.9%, consistent with prior studies.
average time to failure
With a 10% network error rate, the protocol’s local-error threshold ranges from 0.6% to 0.82% across variants; fixing local errors at 0.6% yields a network threshold near 10.0–10.1%.
- 0.6% to 0.82% is the threshold range for local intra-cell errors when the network error rate is fixed at 10%.The range depends on the protocol details.
- 10.0% to 10.1% is the network-error threshold for EXPEDIENT when pm = pg = 0.6% local errors are fixed.This sweep is consistent with the calculations reported for the main-paper setting.
Supplementary Tables
The supplementary tables document modelling probabilities for EXPEDIENT, parameters for STRINGENT, and weighting coefficients for example superoperators.
- Supplementary Table S1 lists probabilities used to model the number of steps required for EXPEDIENT.FRL denotes Failure Reset Level.
- Supplementary Table S2 provides the corresponding STRINGENT parameters for pg = pm = 0.75% and pn = 10%.
- Supplementary Table S3 lists weighting coefficients for three example superoperators.The coefficients are explained in the Supplementary Methods.
Supplementary Note 1: Five qubits per cell
Adding ancillas can improve tolerance to network or intra-cell errors through resource purification and filtering, although optimizing larger-cell protocols is beyond this paper’s scope.
- Four-qubit EXPEDIENT and STRINGENT cells contain one data qubit and three ancillas; adding ancillas can improve performance.
- pn = 0.2 network noise becomes tolerable, rather than pn = 0.1, at the same 0.77% local-error threshold after naive purification extensions.The extensions use additional ancillas for raw-pair creation and double-selection purification.
- STRINGENT+ filters the GHZ resource after coupling it to data and aborts failed filtering before a guaranteed-completion stabilization round.The filter passes in about 92% of cases for the parameter range considered and requires Z-basis measurement.
Supplementary Note 2: Time costs and memory errors
The EXPEDIENT protocol’s probabilistic GHZ construction creates substantial timing variability, while memory errors are estimated to be small over the protocol duration. STRINGENT raises the error threshold but costs roughly five times longer, making the choice depend on active versus passive errors.
- EXPEDIENT timing: EXPEDIENT completes 50% of operations after 57 time steps, 95% after 138, 99% after 195, and 99.9% after 278.The analysis waits for 99% of stabilizers and abandons the remaining 1%.
- Memory errors: A 2-second memory lifetime implies an error rate of about 0.1% over the estimated 2ms EXPEDIENT protocol.This is nearly an order of magnitude below the active gate and measurement error rates considered in the main paper.
- STRINGENT timing: STRINGENT takes about five times longer to achieve the same 99% of complete stabilizers, with a mean duration of 278 time steps.Its higher timing cost is the price paid for an increased error threshold.
- Protocol choice: The better protocol depends on whether active stabilizer-measurement errors or passive memory errors are more severe.The supplementary analysis focuses on EXPEDIENT because it is designed for settings where memory errors matter.
Supplementary Note 3: Physical timescales
Physical operation times vary substantially across system classes. Optical measurement can take about 100µs, remote entanglement may take minutes because of photon loss, and local gates are constrained by interaction strengths and other experimental limits.
- Platform dependence: Operation timescales vary considerably between physical platforms, with some offering substantial room for improvement and others approaching fundamental limits.The relevant operations include measurement, long-range entanglement, and local qubit-qubit gates.
- Measurement: Optical single-shot measurements typically take about 100µs, although trapped Ca ions have achieved 10µs measurement beyond the required fidelity.An NV system has used a 5.5µs window with limited fidelity.
- Remote entanglement: Remote entanglement demonstrations can require minutes because successful heralding often depends on detecting two photons without loss.Photon loss causes repeated failures before eventual entanglement success.
- Local gates: Local conditional-gate speed is limited by the interaction strength between the two qubits and may be further constrained by applied-field intensity.Reported timescales vary from a few microseconds across experimental systems.
Supplementary Methods
The supplementary methods represent noisy stabilizer measurements as mixtures of correct or incorrect ideal projections and Pauli errors. Twirling and circuit symmetries simplify the resulting error weights and make many terms identical.
- Twirling: Twirling makes individual a or b terms in grouped error weights equal, while circuit pairings otherwise create slight qubit-dependent irregularities.The protocol pairs A−B and C−D in Phase 1, then A−C and B−D in Phase 2.
- Superoperator representation: The superoperator acts on four data qubits, with E_e = (ABCD)_e and each operator chosen from {1, σx, σy, σz}.The four operators act on data qubits 1 through 4, while M denotes the reported odd or even outcome.
- Projectors: The perfect projector P^M_ideal represents parity projection in either the Z or X basis, depending on the stabilizer class.The complementary outcome is denoted by M̄.
- Superoperator representation: The real projector combines correct and incorrect ideal projectors with possible Pauli errors weighted by coefficients a and b.The same weight set applies to even and odd reported outcomes in the examples described.
- Example weights: For EXPEDIENT at pn = 0.1 and pm = pg = 0.006, the model gives a 91% chance of a pure correct projection and a 6.2% chance of a pure wrong projection.It also reports a 0.68% chance of a correct projection followed by a single σZ error.
- Example cases: The STRINGENT comparison uses pn = 0.1 and pm = pg = 0.0075, while the monolithic comparison uses pm = pg = 0.09 without network noise.These cases define the parameter settings used for the supplementary comparison.