Mitigating Temporal Fragility in the XY Surface Code

arXiv:2310.17697 · quant-ph · Submitted 2023-10-26 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Mitigating Temporal Fragility in the XY Surface Code".

Mira: Here's the gist of the paper. The gist: A new logical state preparation protocol for the XY surface code mitigates fragile temporal boundaries by using locally entangling qubits into small Greenberger–Horne–Zeilinger-like states prior to stabilizer measurements, leading to a quadratic reduction in fault-configurations and guaranteeing a 50% codecapacity threshold during state preparation for infinitely biased noise Problem with Standard Preparation The standard logical state preparation protocol suffers from fragile temporal boundaries that arise during the process of preparing the logical state +L⟩ by initializing each physical qubit in the +⟩ state and then measuring all stabilizers In this process, half of the stabilizers cannot be used for detecting phase errors, which destroys the symmetry of the XY code, a phenomenon referred to as temporal fragile boundaries As a consequence, the code does not reduce to a repetition code under pure phase noise and the threshold at infinite bias degrades to ∼ 11% Moreover, during state preparation the code distance to phase errors scales as √n where n is the number of physical qubits of the code This fragile boundary-induced degradation also arises during standard approach for logical measurements.

Kai: New State Preparation Protocol The proposed new protocol addresses these issues by entangling physical qubits into local two- and four-body Greenberger–Horne–Zeilinger(GHZ) states before making the stabilizer measurements that place them in the XY-code state This initialization pattern allows for using three-fourths of the stabilizers for correcting phase noise The protocol is described in two steps, beginning with initializing qubits in Bell states as indicated in Fig.

Mira: 2.

Kai: By preparing physical qubits into GHZ states, the initial state becomes a +1 eigenstate of all the X-stabilizers and X-type logical operators.

Title and authors: Kai: We've been talking about how this paper, "Mitigating Temporal Fragility in the XY Surface Code," introduces a new protocol to fix fragile temporal boundaries during logical state preparation.

Mira: That boundary issue is really tricky because standard methods cause half of the stabilizers to become unusable for detecting phase errors, which breaks the symmetry of the XY code.

Lev: And that degradation means when you look at pure phase noise, the threshold drops down to about eleven percent <ref:2310.17697#pg3>.

Kai: This paper proposes initializing physical qubits into small Greenberger–Horne–Zeilinger-like states before measuring stabilizers to get the XY-code state.

Mira: So they use these local entanglements to allow them to use three-fourths of the Y-stabilizers for detecting phase errors, which seems like a big improvement.

Lev: They also show that this initialization pattern makes the initial state a +one eigenstate of all the X-stabilizers and X-type logical operators <ref:2310.17697#pg3>.

Kai: And they prove that in this new procedure, O(√n) high-rate errors along a single lattice boundary can cause a logical failure, leading to an almost quadratic reduction in the number of fault-configurations compared to the standard state-preparation approach <ref:2310.17697#pg1>.

Mira: That quadratic reduction is what they're really highlighting, moving us from something that scales with alpha between three point four one and three point six seven in the standard preparation protocol <ref:2310.17697#pg2>.

Lev: And specifically for pure phase noise, their scheme is able to mitigate the degrading effect of fragile boundaries <ref:2310.17697#pg2>.

Kai: They also have a strong result that the code becomes equivalent to a repetition code for high-rate errors, which they prove using distance equal to square root n <ref:2310.17697#pg1>.

Mira: So, when we think about the naive question of how this helps us in practice, it seems like it gives us a much more stable setup for preparing the state itself <ref:2310.17697#pg3>.

Lev: For running on real hardware, this means that the overhead associated with state preparation is reduced because of this better error correction scaling <ref:2310.17697#pg4>.

The paper's summary: Kai: The paper really summarizes that they address the problem of fragile temporal boundaries by entangling qubits into GHZ states before measurement to place them in the XY-code state.

Mira: They detail two steps for this protocol: first, initializing qubits into Bell states and creating GHZ states like ϕ4⟩ around Y-stabilizer plaquettes <ref:2310.17697#pg3>.

Lev: Then, in step two, they measure all the stabilizers, and because of that initial entanglement, they can detect errors on half of the Y-stabilizers <ref:2310.17697#pg3>.

Kai: The summary emphasizes that this allows them to correct Z errors on all bulk qubits and decode Z errors on data qubits at the temporal boundary as a single repetition code of length d <ref:2310.17697#pg3>.

Mira: So, the core implication is that this method solves the problem by fundamentally changing how we initialize the state to make it much more resilient <ref:2310.17697#pg4>.

Lev: From an engineering standpoint, this means we can use short-depth circuits for Bell-state preparation that are compatible with conventional surface code layouts <ref:2310.17697#pg2>.

Kai: The paper's summary also points out that they discuss how this state-preparation scheme can be inverted for fragile-boundary-mitigated logical state measurement <ref:2310.17697#pg2>.

Mira: That’s really interesting because it opens up possibilities for AI systems to perform logical measurements in noisy environments with significantly reduced errors compared to standard approaches <ref:2310.17697#pg4>.

The paper's improvements: Kai: The main improvements they suggest are the quadratic reduction in fault-configurations due to pure Z errors, which is O(√n) scaling instead of O(α √n) scaling where alpha is between three point four one and three point six seven <ref:2310.17697#pg2>.

Mira: They also show a quantitative improvement on the threshold to pure Z noise, guaranteeing a fifty percent code capacity threshold during state preparation for infinitely biased noise <ref:2310.17697#pg4>.

Lev: And they provide the numerical evidence that shows this works even under fault-tolerant setting where some bit-flip errors and measurement errors are present <ref:2310.17697#pg2>.

Kai: Furthermore, the simulations show that at experimentally realistic high bias of eta equals one hundred four the threshold for the total error rate with this scheme is fifteen point one percent compared to ten point two percent for the standard approach <ref:2310.17697#pg4>.

Mira: And they also found that they can suppress logical error rates by a factor of approximately two for certain noise regimes, as the paper obtains Lambda equals two point three(one) which is nearly twice of Lambda equals one point two(one) for the standard protocol <ref:2310.17697#pg4>.

Lev: The limitation they flag is that their analytical results are limited to the case of pure phase noise <ref:2310.17697#pg2>.

Kai: So, what this means for us is that while the paper shows strong improvements, we have to be careful because those specific analytical results only apply when you're dealing with pure phase noise <ref:2310.17697#pg2>.

Conclusion: Kai: So, in conclusion, this paper on "Mitigating Temporal Fragility in the XY Surface Code" successfully mitigates the degrading effect of fragile boundaries through local GHZ states for improved initialization <ref:2310.17697#pg4>.

Mira: It confirms that temporal fragility can be addressed in the XY surface code by employing local GHZ states for improved initialization <ref:2310.17697#pg3>.

Lev: For us, this means we have a scheme that handles Z error correction and shows it works at experimentally realistic noise biases like eta equals one hundred four <ref:2310.17697#pg4>.

Kai: The key finding is that there are O(√n) least-weight fault-configurations due to pure Z errors, which is nearly quadratic improvement over the standard state-preparation approach where alpha is between three point four one and three point six seven <ref:2310.17697#pg2>.

Mira: It also confirms that the code reduces to a single repetition code for phase errors with distance equal to square root n <ref:2310.17697#pg4>.

Lev: And they confirm that there is a fifty percent threshold against phase-noise in the state preparation process for infinitely biased noise <ref:2310.17697#pg4>.

Kai: We're looking at how this impacts the XY surface code by employing local GHZ states to fix temporal fragility <ref:2310.17697#pg4>.

Lev: And we have this method that addresses the Z error correction and shows it works at experimentally realistic noise biases like eta equals one hundred four <ref:2310.17697#pg4>.

Kai: That's where we are right now, looking at how this impacts the XY surface code by employing local GHZ states to fix temporal fragility <ref:2310.17697#pg4>.

Pei-Kai Tsai, Yue Wu, Shruti Puri

Yale University

quant-ph

Submitted: 2023-10-26

Updated: 2024-08-14

Comments: 9 pages, 9 figures, v2

Journal ref: Phys. Rev. X 14, 031003 (2024)

DOI: 10.1103/PhysRevX.14.031003

Code: https://github.com/yuewuo/QEC-Playground

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 90/100

The gist: The gist: A new logical state preparation protocol for the XY surface code mitigates fragile temporal boundaries by using locally entangling qubits into small Greenberger–Horne–Zeilinger-like

Key concepts

Temporal Fragile Boundaries: These are issues in the standard state preparation where initializing qubits in |+⟩ and measuring stabilizers leaves half of the stabilizers unusable for detecting phase errors. This destroys the symmetry of the XY code, causing performance degradation and a lower fault-tolerance threshold.
Greenberger–Horne–Zeilinger (GHZ) States: The new protocol entangles physical qubits into specific GHZ states before stabilizer measurements. These entangled states act as +1 eigenstates for certain X-stabilizers, allowing the scheme to utilize more stabilizers for error correction and improve fault tolerance.
Code Capacity Threshold: This is the maximum noise level (error rate) a quantum code can tolerate while still reliably protecting the logical information. The new protocol guarantees a 50% threshold against pure Z noise during state preparation, which is significantly better than standard methods.

Terminology

Decoder — 1× in this paper · explained in 2 episodes

Summary

The gist: A new logical state preparation protocol for the XY surface code mitigates fragile temporal boundaries by using locally entangling qubits into small Greenberger–Horne–Zeilinger-like states prior to stabilizer measurements, leading to a quadratic reduction in fault-configurations and guaranteeing a 50% codecapacity threshold during state preparation for infinitely biased noise

Problem with Standard Preparation

The standard logical state preparation protocol suffers from fragile temporal boundaries that arise during the process of preparing the logical state +L⟩ by initializing each physical qubit in the +⟩ state and then measuring all stabilizers In this process, half of the stabilizers cannot be used for detecting phase errors, which destroys the symmetry of the XY code, a phenomenon referred to as temporal fragile boundaries As a consequence, the code does not reduce to a repetition code under pure phase noise and the threshold at infinite bias degrades to ∼ 11% Moreover, during state preparation the code distance to phase errors scales as √n where n is the number of physical qubits of the code This fragile boundary-induced degradation also arises during standard approach for logical measurements.

New State Preparation Protocol

The proposed new protocol addresses these issues by entangling physical qubits into local two- and four-body Greenberger–Horne–Zeilinger(GHZ) states before making the stabilizer measurements that place them in the XY-code state This initialization pattern allows for using three-fourths of the stabilizers for correcting phase noise The protocol is described in two steps, beginning with initializing qubits in Bell states as indicated in Fig. 2. By preparing physical qubits into GHZ states, the initial state becomes a +1 eigenstate of all the X-stabilizers and X-type logical operators. Importantly, ϕ4⟩ and ϕ2⟩ are +1 eigenstates of Y⊗4 and Y⊗2, respectively. Consequently, in the absence of errors, the measurement outcomes of the marked Y-stabilizers must be +1. This allows half of the Y-stabilizers to be used to detect errors in the new scheme.

Performance Gains and Theoretical Guarantees

The new protocol guarantees that Z errors on all the bulk qubits are correctable (part I) and Z errors on data qubits at the temporal boundary can be decoded as a single repetition code of length d (part II) Corollary 1 states that there are O(2√n) least-weight fault-configurations due to pure Z errors, which is nearly quadratic improvement over the least-weight fault-configurations in the standard state-preparation approach where 3.41 ≤ α ≤ 3.67 Furthermore, Corollary 2 confirms that the threshold to pure Z noise is 50% during state preparation for infinitely biased noise.

Numerical Simulations and Practical Performance

Numerical simulations examine the performance of the new scheme under fault-tolerant setting where some bit-flip errors and measurement errors are present Benchmarking under ideal measurements (pm = 0) with only pure Z noise shows that the adapted Tuckett decoder results in a 49.1(2)% threshold for our state-preparation scheme and 10.1(1)% threshold for the standard scheme, which align with analytical predictions. For experimentally realistic high bias η = 104, the threshold for the total error rate with our scheme is 15.1(2)% in comparison to that of 10.2(1)% with the standard approach. The logical error rate for our scheme is several orders of magnitude smaller than with the standard preparation scheme for fixed p.

Implementation and Future Work

The protocol can be inverted for fragile-boundary-mitigated logical state measurement Preparing the GHZ states requires a short-depth circuit compatible with conventional surface code layout, and an alternative circuit is provided that introduces minimum additional noise while eliminating first-order error correlations on data qubits caused by Z errors on the ancilla. Future work should consider a full circuit-level simulation of our scheme with additional modifications to mitigate the fragile spatial boundaries There is also room for improving the decoder by combining the Tuckett decoder with belief-propagation or using a hypergraph decoder. The problem of temporal fragile boundary appears in other codes like the tailored XZZX surface code, where local entangled states may allow fixing more stabilizers and gaining information about errors in the first round The paper also discusses how to use measurement outcomes to infer logical measurements by summing over disjoint XX and X measurements.

Conclusion

The proposed protocol successfully mitigates the degrading effect of fragile boundaries, demonstrating better threshold performance and faster suppression of logical error rates compared to the standard state-preparation approach. The scheme confirms that temporal fragility can be addressed in the XY surface code by employing local GHZ states for improved initialization.

How it works

Standard Logical State Preparation involves initializing each physical qubit in the +⟩ state followed by a measurement of all code stabilizers. This results in half of the Y-stabilizers being unusable for detecting Z errors, leading to a degradation of the XY code performance.
The New Protocol involves two steps: first, initializing qubits into Bell states such that four physical qubits around the Y-stabilizer plaquettes are entangled into a GHZ state ϕ4⟩ = √1/2+⟩⊗4 + -⟩⊗4, and each pair involved in the two-body Y-stabilizer is entangled into the Bell state ϕ2⟩ = √1/2(++⟩ − --−) and the remaining qubits are prepared in +⟩.
In step 2, all stabilizers are measured; because of the initial entanglement, this measurement allows for error detection on half of the Y-stabilizers. The analysis shows that Z errors on all bulk qubits are correctable and Z errors on data qubits at the temporal boundary can be decoded as a single repetition code REP(d) of length d.

Key Findings

**: In the new protocol, there are O(2√n) least-weight fault-configurations due to pure Z errors, which is nearly quadratic improvement over the standard state-preparation approach where 3.41 ≤ α ≤ 3.67. This leads to a 50% threshold against phase-noise in the state preparation process. The code reduces to a single repetition code for phase errors with distance √n. The threshold for Z errors in the quantum code is also 50% from which Corollary 2 follows. The suppression factor Λ for p = 0.05 is nearly twice that of the standard protocol, indicating much faster suppression in logical error rates as larger codes are used with our protocol. The threshold for the total error rate with our scheme at η = 104 is 15.1(2)% compared to 10.2(1)% for the standard approach. The paper confirms that the scheme can reduce additional state-preparation errors due to temporal fragile boundaries. The protocol can be inverted for fragile-boundary-mitigated logical measurements.

Improvements for AI systems

This protocol can be inverted for fragile-boundary-mitigated logical state measurement, allowing AI systems to perform logical measurements in noisy environments with significantly reduced errors compared to standard approaches.
AI systems can achieve a 50% threshold against phase-noise is guaranteed in the state preparation process, substantially improving robustness during initial quantum computation steps.
The system can utilize a reduction of O(2√n) fault-configurations due to pure Z errors, which is almost quadratic reduction compared to the least-weight fault-configurations in the standard state-preparation approach.
For finite measurement error, the scheme maintains performance, as at high bias and realistic noise levels (η = 104), it achieves a threshold of 15.1(2)% in comparison to that of 10.2(1)% with the standard approach.
The system can suppress logical error rates by a factor of approximately two for certain noise regimes, as the paper obtains Λ = 2.3(1) which is nearly twice of Λ = 1.2(1) for the standard protocol.

Abstract

An important outstanding challenge that must be overcome in order to fully utilize the XY surface code for correcting biased Pauli noise is the phenomena of fragile temporal boundaries that arise during the standard logical state preparation and measurement protocols. To address this challenge we propose a new logical state preparation protocol based on locally entangling qubits into small Greenberger-Horne-Zeilinger-like states prior to making the stabilizer measurements that place them in the XY-code state. We prove that in this new procedure O(sqrt n) high-rate errors along a single lattice boundary can cause a logical failure, leading to an almost quadratic reduction in the number of fault-configurations compared to the standard state-preparation approach. Moreover, the code becomes equivalent to a repetition code for high-rate errors, guaranteeing a 50% code-capacity threshold during state preparation for infinitely biased noise. With a simple matching decoder we confirm that our preparation protocol outperforms the standard one in terms of both threshold and logical error rate in the fault-tolerant regime where measurements are unreliable and at experimentally realistic biases. We also discuss how our state-preparation protocol can be inverted for similar fragile-boundary-mitigated logical-state measurement.

Sources

Belief propagation as a partial decoder