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Surface codes: Towards practical large-scale quantum computation

Austin G. Fowler, Matteo Mariantoni, John M. Martinis, Andrew N. Cleland

arXiv:1208.0928v2quant-ph

TL;DR

Quantum errors threaten reliable computation, motivating architectures that can detect and tolerate them. This paper introduces surface-code computing and finds that logical states remain intact below a 0.57% physical error threshold.

  • Problem

    Quantum states are delicate and unintended errors remain a major challenge for reliable quantum computing.

  • Method

    The paper develops surface-code computing through stabilizers, logical qubits, braids, universal gates, and ancilla-state distillation.

  • Results

    0.57% is the reported per-step physical-qubit error threshold below which surface codes preserve logical-state integrity.

  • Takeaways & Limitations

    Surface codes combine substantial error tolerance with a two-dimensional nearest-neighbor layout, making them a realistic solid-state quantum-computing architecture.

  • Takeaways & Limitations

    With only two X and two Z boundaries, the described array provides just one logical qubit regardless of its size.

Abstract

from arXiv · show

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

I. BACKGROUND · II. INTRODUCTION

The paper motivates surface codes as a route to fault-tolerant quantum computation, offering high error tolerance at the cost of substantial physical-qubit overhead. It introduces stabilizer-based error detection, showing how compatible multi-qubit measurements support logical qubits while preserving quantum information.

  • I. BACKGROUND: About 1% per operation is the surface code’s error tolerance, versus about 2 × 10^-5 for Steane and Bacon-Shor codes on two-dimensional nearest-neighbor lattices.The latter therefore require error rates three orders of magnitude lower than the surface code.
  • I. BACKGROUND: About 10^3 to 10^4 physical qubits are needed for a reasonably fault-tolerant surface-code logical qubit, despite a minimum of thirteen physical qubits.The required number depends strongly on the physical-qubit error rate.
  • I. BACKGROUND: About one billion physical qubits operating for about one day are estimated for factoring an N = 2,000-bit number under the paper’s assumptions.The estimate includes about 4,000 computational logical qubits and 2.2 × 10^12 |AL⟩ states.
  • I. BACKGROUND: About 130 million physical qubits can result from improving the overall physical-qubit error rate by about a factor of ten, while execution time remains unchanged.The execution time is set by modular exponentiation and can only be reduced by reducing measurement time.
  • II. INTRODUCTION: Surface-code logical qubits use entangled physical qubits, repeated CNOT operations, and subsequent measurements for error correction and error detection.The resulting logical qubit performs far better than its underlying physical qubits.
  • II. INTRODUCTION: The surface code emphasizes detecting errors that affect measurement outcomes, allowing identified errors to be undone in classical software rather than correcting every physical error directly.A Z measurement requires correction for an X error but not for a Z error.
  • II. INTRODUCTION: Measuring XaXb and ZaZb on two qubits avoids the incompatibility of sequential single-qubit X and Z measurements because the two operators commute.Their simultaneous eigenstates are the four Bell states.
  • II. INTRODUCTION: Two-qubit stabilizer measurements signal that an X or Z error occurred, but identical measurement changes can arise from different errors, so uniquely identifying errors requires a more complex system such as the surface code.This limitation motivates extending the two-qubit construction to larger stabilizer systems.

III. THE SURFACE CODE … VI. LOGICAL OPERATORS

The surface code uses a two-dimensional array of data and measurement qubits to detect and localize errors through stabilizer measurements. Its unconstrained degrees of freedom form logical qubits manipulated by anticommuting operator chains.

  • III. THE SURFACE CODE: The surface code is implemented on a two-dimensional array containing data qubits that store computational states and measurement qubits that stabilize them.Each data qubit couples to two measure-Z and two measure-X qubits, while each measurement qubit couples to four data qubits.
  • IV. QUIESCENT STATE OF THE SURFACE CODE: A complete stabilizer cycle initializes each measurement qubit, applies four CNOT operations, and performs a projective measurement.For measure-Z qubits, the four neighboring data qubits control the CNOTs targeting the measurement qubit.
  • IV. QUIESCENT STATE OF THE SURFACE CODE: The cycle projects the data qubits into one of 2^N quiescent states, which remain unchanged without errors because all stabilizers commute.For N = 38 measure qubits, the array has 2^38 ≈ 3×10^11 possible quiescent states.
  • V. SINGLE QUBIT ERRORS: Single-qubit errors produce characteristic sign changes in neighboring syndrome measurements, allowing phase-flip, bit-flip, and combined errors to be detected and localized.A phase-flip on one data qubit changes the outcomes of its two adjacent measure-X qubits, while leaving adjacent measure-Z outcomes unchanged.
  • V. SINGLE QUBIT ERRORS: When errors are sufficiently rare, spatial and temporal syndrome signals can be matched to identify errors with very high probability and corrected in classical software.Software can track the affected qubits and adjust subsequent syndrome-measurement signs instead of applying potentially imperfect physical corrections.
  • VI. LOGICAL OPERATORS: Because the stabilizers are not always complete, additional degrees of freedom remain and can be encoded as logical operators acting without changing stabilizer measurements.In the illustrated array, 41 data qubits and 40 measure qubits leave two unconstrained degrees of freedom.
  • VI. LOGICAL OPERATORS: Logical X and Z operators are chains of paired single-qubit operations crossing the array between matching boundaries, and alternative chains differ only by stabilizer products.There is only one linearly independent logical X operator and one linearly independent logical Z operator for the illustrated array.
  • VI. LOGICAL OPERATORS: The logical X, Y, and Z operators obey the physical-qubit commutation relations, making the two-dimensional array a logical qubit; n logical qubits give |qL⟩ dimension 2^n.Logical X and Z anticommute because their chains intersect on one data qubit.

VII. ERROR DETECTION · A. Statistical model for the logical error rate

The section models surface-code logical errors arising from physical, measurement, initialization, gate, and decoding errors, establishing a threshold below which increasing array distance suppresses logical errors. It then explains this scaling statistically and uses it to estimate spatial and temporal resources for target logical error rates.

  • VII. ERROR DETECTION: Surface-code error correction handles physical, measurement, initialization, and gate errors when errors during each cycle can be identified or tracked up to stabilizer-equivalent chains.Edmonds’ minimum-weight perfect-matching algorithm provides automated decoding, mapping stabilizer changes to likely physical errors.
  • VII. ERROR DETECTION: For p < pth, the logical error rate PL falls exponentially with distance d, whereas for p > pth it increases with d; the simulated threshold is pth = 0.57%.The threshold corresponds to roughly a 4% error rate for the entire surface-code measurement cycle.
  • VII. ERROR DETECTION: Below threshold, the simulations find PL ∼ p^de, with the effective error dimension determined by the minimum number of physical bit- or phase-flips needed to form a logical operator.For even d, the effective dimension is rounded down to de = d/2.
  • A. Statistical model for the logical error rate: In the statistical model, indistinguishable error reports can arise from complementary chains, but shorter chains dominate because a two-error event scales as p^2 while a three-error event scales as p^3.The estimate considers data-qubit errors and ignores, for example, CNOT errors; multiplying by d accounts for d independent rows.
  • A. Statistical model for the logical error rate: The statistical prediction scales similarly to the simulations and estimates the physical-qubit cost per logical qubit as nq = (2d−1)^2.The estimate uses total data and measurement qubits and examines nq versus p/pth for several target PL values.
  • A. Statistical model for the logical error rate: The required physical-qubit count rises rapidly as p approaches pth, making gate fidelity above about 99.9%—p ≲ 10^-3—a useful target for achieving practical logical error rates.The scaling relations are used to estimate resources for desired PL values and to assess requirements for Shor’s algorithm.
  • A. Statistical model for the logical error rate: Temporal measurement errors follow the same misidentification scaling as spatial errors, so maintaining the minimum logical error rate requires roughly the same distance d in time and space when pM is comparable to p.Here pM denotes the measurement-error rate per surface-code cycle, and temporal distance is measured in complete cycles.

B. Logical error rate for different error classes · VIII. CREATING LOGICAL QUBITS · X2 XL

The paper distinguishes logical-error sensitivity across physical error classes and explains how surface-code boundaries and holes create logical qubits. It further develops double-cut qubits, locality constraints, and qubit-type requirements for topological CNOT operations.

  • B. Logical error rate for different error classes: Class-0 errors affect data-qubit idle operations, class-1 errors affect measure-qubit operations, and class-2 errors arise from CNOT operations.Class-0 errors have four opportunities per surface-code cycle; class-1 errors include initialization, measurement, and Hadamard operations, with some idle errors having no impact.
  • B. Logical error rate for different error classes: Logical error rates are least sensitive to class-1 errors, more sensitive to class-0 errors, and most sensitive to class-2 errors, with different thresholds for each class.When errors occur concurrently at PL = 0.02, class-0 and class-2 errors have roughly equal impact, while class-1 errors are roughly five times less influential.
  • B. Logical error rate for different error classes: 0.57% is the per-step physical-qubit error threshold, and increasing array distance d lowers logical error rates when the physical error rate remains below threshold.Physical errors are continuously detected through stabilizer measurements and accounted for in software.
  • VIII. CREATING LOGICAL QUBITS: Turning off a single measure-Z qubit creates a Z-cut hole with two additional degrees of freedom, enabling anti-commuting logical X and Z operators for one logical qubit.The logical X chain connects the array’s outer X boundary to the hole’s internal X boundary, while the logical Z operator loops around the hole.
  • X2 XL: A Z-cut qubit requires at least one X boundary and has distance d = 3 in the illustrated configuration, while an analogous X-cut qubit requires at least one Z boundary.The Z-cut chain and loop share one data qubit, causing anti-commutation; moving the hole farther from the boundary can increase the distance.
  • X2 XL: Turning off two measure-Z qubits creates a double Z-cut qubit with four additional degrees of freedom, but locality requires restricting the operator set to manipulate one effective two-level qubit.Double-cut constructions avoid operators reaching array boundaries, while local logical operators support parallel gate operations.
  • X2 XL: Z-cut and X-cut logical qubits are both needed for topological braids implementing logical CNOT, although same-type CNOTs can use the opposite type as an intermediary.Only braids between mixed qubit types directly provide the needed functionality.
  • X2 XL: Small double-cut logical qubits have distance d = 3 and are relatively fault-intolerant, whereas larger-distance constructions are described as significantly more fault-tolerant.For the illustrated Z-cut qubit, the Z loop uses four data-qubit Z operators and the X chain uses three data-qubit X operators.

IX. SOFTWARE-IMPLEMENTED ˆZL AND ˆ … XII. LARGER LOGICAL QUBITS

The surface code handles logical Pauli operators in classical software, supports easy and difficult initialization and measurement procedures, and preserves fault tolerance during stabilizer manipulations. Enlarging and spacing logical-qubit holes lengthens logical operators and improves error tolerance.

  • IX. SOFTWARE-IMPLEMENTED ˆZL AND ˆ: Logical X_L and Z_L operations are handled entirely by classical control software rather than physically applied to the logical qubits.The software commutes these operators through circuit gates until they cancel or correct a measurement outcome.
  • IX. SOFTWARE-IMPLEMENTED ˆZL AND ˆ: At terminal measurements, pending operators modify outcomes by reversing the sign of an affected X_L measurement while leaving an X_L correction on another qubit unaffected.In the example, M_X = ±1 becomes −M_X = ∓1 for qubit 1, while qubit 2 is unchanged.
  • X. LOGICAL QUBIT INITIALIZATION AND MEASUREMENT: Logical qubits use “easy” initialization and measurement in matching logical eigenbases, while opposite-basis operations require more involved procedures.The difficult procedures can involve turning stabilizers on or off, modifying terminal measurements, measuring individual data qubits, and resetting them.
  • A. Initialization.: For an X-cut qubit, easy initialization uses an X_L eigenstate, whereas difficult Z_L-basis initialization can directly create |g_L⟩ or |e_L⟩ through stabilizer and data-qubit manipulations.Z-cut initialization is analogous: Z_L eigenstates are easy, while X_L eigenstates require the difficult procedure.
  • B. Measurement: Easy measurement turns on stabilizers in the qubit holes, while difficult measurement uses a reverse-style procedure to measure the logical qubit in the opposite basis.The logical outcome is obtained fault-tolerantly from stabilizer outcomes or, for difficult measurements, from measured isolated data qubits.
  • XI. ERRORS DURING STABILIZER MANIPULATIONS: Careful measurements before and after stabilizer manipulations, including modified error-location algorithms, preserve the same fault tolerance as the rest of the surface code.The procedures address errors on isolated and partly stabilized data qubits while maintaining error detection.
  • XII. LARGER LOGICAL QUBITS: Increasing hole size and spacing improves logical error tolerance by increasing the number of physical-qubit operators in the logical X_L and Z_L chains.For a five-stabilizer Z-cut hole, the Z_L loop grows from four to eight physical-qubit operators, while separating holes lengthens the X_L chain.
  • XII. LARGER LOGICAL QUBITS: A larger Z-cut qubit is initialized from the product of four Z-stabilizer outcomes and measured along Z_L by the stable product Z_s1Z_s2Z_s3Z_s4 = ±1.Difficult X_L measurement multiplies the X eigenvalues of isolated data qubits, then resets those qubits and restores all stabilizers.

XIII. MOVING QUBITS · A. One-cell logical qubit move · B. Byproduct operators

The surface code moves logical qubit holes by coordinating physical stabilizer operations with unitary transformations of the logical operators. Measurement-dependent sign changes produce byproduct operators, which are tracked in software and applied only when the logical qubit is measured.

  • XIII. MOVING QUBITS: Logical-qubit motion and braiding provide a central surface-code function, with braiding implementing a logical CNOT.A logical qubit hole is moved between the two holes of another logical qubit to braid them together.
  • XIII. MOVING QUBITS: A logical-qubit move is described in two stages: physical surface-code operations and transformations of the logical X_L and Z_L operators.The logical-operator transformations are expressed in the Heisenberg representation.
  • XIII. MOVING QUBITS: Physical moving and braiding operations use projective measurements and are nonunitary, whereas the corresponding logical-operator transformations are unitary.The discussion assumes no errors occur on physical qubits or during measurements.
  • A. One-cell logical qubit move: Two complete surface-code cycles move a Z-cut logical qubit hole down by one cell while redefining its logical operators.The procedure turns off the lower Z stabilizer, measures data qubit 6, waits d surface-code cycles, and turns the original stabilizer back on.
  • A. One-cell logical qubit move: After the move, the new logical operators still share one data qubit and commute with all stabilizers, so they continue to define a logical qubit.The move uses operations similar to those in the easy and difficult initializations, so its error processes are treated similarly.
  • B. Byproduct operators: Measurement outcomes can change the signs of the transformed logical X_L and Z_L operators, creating bit- and phase-flip byproduct operators.The X_L sign depends on the X6 outcome, while the Z_L sign depends on the product of the two Z-stabilizer outcomes.
  • B. Byproduct operators: Byproduct operators are tracked rather than applied as explicit gates, and their corrections are incorporated when the logical qubit is measured.The relevant measurement sign is reversed according to the tracked opposite logical byproduct operator.

C. Multi-cell logical qubit move … B. Braiding two qubits

Multi-cell moves translate logical-qubit holes across arbitrarily long strips without increasing the clock-cycle cost, while braiding holes produces operator transformations equivalent to a logical CNOT. The sections also describe move-error handling and the operator-level mechanism underlying this equivalence.

  • C. Multi-cell logical qubit move: Multi-cell moves translate a logical-qubit hole over an unlimited number of cells in the same number of surface code clock cycles as a one-cell move.The move extends the one-cell procedure across a contiguous strip of cells.
  • s,j. (b) Extension of ˆZL to ˆZe: The multi-cell procedure extends logical operators through intervening stabilizers, measures isolated data qubits along X, restores stabilizers, and waits d−1 additional cycles.For X-cut holes, the analogous procedure exchanges X and Z stabilizers and measurements.
  • D. Errors during move transformations: Move errors are handled by detecting and localizing Z errors on data qubits bordering the cut, while waiting d cycles distinguishes persistent data errors from stabilizer-measurement errors.X errors on isolated data qubits have no impact because the single-qubit X measurement erases them.
  • XIV. THE BRAIDING TRANSFORMATION AND THE LOGICAL CNOT: A braid consists of two multi-cell moves that carry one hole around a closed loop, and the transformation can entangle two logical qubits equivalently to a logical CNOT.The shifted hole returns to its original location after the two moves.
  • A. Braid transformation of one logical qubit: For a single Z-cut qubit, braiding adds a closed X-operator loop to X_L, whereas Z_L returns to its original loop apart from stabilizer-induced sign changes.The differing transformations are identified as the key to the braid’s CNOT action.
  • B. Braiding two qubits: When a Z-cut qubit braids through an X-cut qubit, X_L1 ⊗ I_L2 transforms to X_L1 ⊗ X_L2, while closed-loop logical operators remain unchanged.Open chains linking holes produce loops around the other hole, whereas closed loops do not interact with it.
  • B. Braiding two qubits: The two braid moves generate sign changes and corresponding byproduct logical operators on both the displaced first qubit and the stationary second qubit.The byproducts include X_L1, Z_L1, X_L2, and Z_L2 operators.

C. The CNOT gate

The section defines the CNOT by its control-dependent target flip and describes Schrödinger- and Heisenberg-picture validation methods. It shows that the braid satisfies the required CNOT transformations, while only the logical-state transformation remains unitary.

  • C. The CNOT gate: The CNOT leaves the target unchanged for control |g⟩ and applies an X bit-flip for control |e⟩.It is a fundamental two-qubit gate with distinct control and target qubits.
  • C. The CNOT gate: A Schrödinger-picture test runs the CNOT on all four basis states and measures each output in all four basis states, producing sixteen experiments.The results are compared with the CNOT matrix to verify correct implementation.
  • C. The CNOT gate: Heisenberg-picture validation requires checking four operator transformations rather than all sixteen outer products of I, X, Y, and Z.The remaining twelve relations are trivial or derivable from the four checked transformations, using CNOT unitarity and Y = Z X.
  • C. The CNOT gate: The four transformations required to validate a CNOT are exactly those obtained for the braid, establishing that a braid is a CNOT.This identifies the braid operation with the CNOT at the level of the logical transformations.
  • C. The CNOT gate: The complete braid, including physical data-qubit measurements, is nonunitary, whereas its transformation of the logical state |q_L⟩ is unitary.The stabilized physical state |Q⟩ undergoes a nonunitary transformation, but the logical-state transformation remains unitary.

D. CNOT between two Z-cut qubits … XVI. SINGLE QUBIT ˆSL AND ˆTL OPERATORS

The paper extends braid-based CNOTs to same-cut qubits and multi-target operations, then implements logical Hadamard, S, and T gates using surface-code patches, ancilla states, measurements, and software-tracked byproducts. These constructions complete the required logical-gate set while preserving error correction under stated code-distance and physical-error conditions.

  • D. CNOT between two Z-cut qubits: A Z-cut-to-Z-cut CNOT uses an ancillary X-cut qubit and three braid-generated logical CNOTs, with the target-in state transferred to target-out.Measurement outcomes determine whether an X_L correction is applied to the target-out qubit.
  • 2. If instead the ˆZ measurement of the second qubit: The circuit implements the computational-basis mappings |gg⟩→|gg⟩, |ge⟩→|ge⟩, |eg⟩→|ee⟩, and |ee⟩→|eg⟩.The |eg⟩ case produces |ee⟩ regardless of the measured ancilla outcome after the prescribed correction.
  • E. Single-control, multi-target CNOTs: An analogous construction performs CNOTs between two X-cut qubits using an intermediate Z-cut ancilla, while single-control, multi-target CNOTs require the same number of surface-code cycles as same-cut CNOTs.The multi-target circuit uses an ancillary Z-cut qubit between X-cut control and target qubits.
  • XV. THE HADAMARD TRANSFORMATION: A logical Hadamard isolates a qubit patch, applies physical Hadamards that exchange X_L and Z_L, and realigns stabilizers through data-measure-qubit swaps and hole repositioning.The example begins with d = 7 logical qubits and uses stabilizer operations to reconnect the transformed patch to the array.
  • XV. THE HADAMARD TRANSFORMATION: Hadamard error correction remains effective when the patch distance d is sufficiently large and physical Hadamard errors are sufficiently rare.The number of repeated stabilizer measurements depends on d; the stated repeat counts are specific to the d = 7 geometry, while larger-distance codes require more repetitions.
  • XVI. SINGLE QUBIT ˆSL AND ˆTL OPERATORS: The T gate uses a |A_L⟩ ancilla and a Z_L measurement: MZ = +1 yields T_L|ψ_L⟩, while MZ = −1 yields a correctable T†_L branch.Approximately half the circuit runs succeed directly, and the other branch is corrected with S_L, leaving software-tracked byproducts.
  • XVI. SINGLE QUBIT ˆSL AND ˆTL OPERATORS: The logical S gate is deterministic: interacting an input with a |Y_L⟩ ancilla through two controlled logical CNOTs and two logical Hadamards produces S_L|ψ_L⟩.The same circuit can produce S†_L with a Z_L byproduct, which is tracked in software.
  • XVI. SINGLE QUBIT ˆSL AND ˆTL OPERATORS: High-fidelity S_L and T_L operations rely on state injection into short qubits, expansion to standard distance d, and distillation of the resulting imperfect logical ancilla states.The circuits also define how X_L and Z_L byproducts commute through S_L and T_L and are handled by classical control software.

B. Short qubits and state distillation

Short qubits enable state injection for the |YL⟩ and |AL⟩ ancilla states, whose precision is then improved through probabilistic distillation. The |YL⟩ and |AL⟩ protocols suppress errors rapidly while requiring relatively small temporal overhead or repeat rates.

  • Short qubits and state injection: A short qubit reduces the logical X_L chain to one qubit, enabling state injection while limiting accumulated errors by minimizing its duration.The short qubit is error-prone, so it should remain small for as few surface code cycles as possible.
  • State distillation: Distillation is required because state injection cannot achieve error rates below 10^-14, whereas |YL⟩ and |AL⟩ states can be made significantly more precise.Distillation probabilistically purifies imperfect input states through repeated executions of a particular logic circuit.
  • Distillation overhead: 74p9: two |YL⟩ distillation cycles require 49 approximate input states and produce exponentially reduced output error; at most eight cycles are needed for |AL⟩ distillation.The temporal overhead is relatively small because the surface code implements these circuits efficiently.
  • |YL⟩ distillation: 7p3 ≪ p: one |YL⟩ distillation cycle reduces input X_L or Z_L error probability p to 7p3, succeeding with probability 1 − 7p.If insufficient accuracy is achieved, additional distillation cycles can be added.
  • |AL⟩ distillation: 35p3: |AL⟩ distillation reduces errors containing a Z_L component from probability p to 35p3, succeeds with probability 1 − 15p, and for p of order 1% is rerun about one-sixth of attempts.Additional distillation cycles improve the output exponentially.

XVII. PHYSICAL IMPLEMENTATIONS

The surface code is motivated by realistic physical implementation, with requirements on processing speed, gate duration, coherence, and fidelity. Superconducting circuits emerge as a leading candidate, although integration with classical logic and achieving sufficient fidelities remain challenging.

  • Motivation: The practical implementation discussion follows a largely theoretical treatment intended to support realistic and practical quantum-computer development.The article notes that the surface code can implement the gates required for algorithms such as Shor’s and Grover’s search.
  • Physical requirements: Surface code implementation requires physical gates lasting 10-100 ns, coherence times of at least 1-10 µs, and minimum gate fidelities of 99%.These requirements follow from applying error-detection rounds at roughly 10^6 to 10^7 Hz alongside classical processor clock rates near 3 GHz.
  • Physical requirements: High-speed classical interconnects make qubit spacing of order 1 µm or less difficult, whereas tens to hundreds of µm should be more straightforward.The challenge arises from tightly intermingling classical and quantum logic at each qubit.
  • Physical requirements: A 10^4 × 10^4 physical-qubit array with 100 µm spacing would occupy about 1 × 1 m^2, suggesting that large qubits may be manageable.The passage presents this array size as perhaps manageable rather than unrealistic.
  • Candidate implementations: Superconducting circuits appear among the best candidates because their physical and operating parameters fall within the discussed ranges.Major remaining challenges include sufficient gate and measurement fidelities and tight integration with classical logic.

Appendix A: Notation · 8. The ˆS gate is another ˆZ-axis rotation, represented · Appendix B: ˆZ and ˆ X stabilizer circuits

The appendices define the article’s operator and gate conventions, then work through a simplified surface-code stabilizer circuit. The circuit projects arbitrary two-qubit inputs onto simultaneous eigenstates of ˆXa ˆXb and ˆZa ˆZb, provided the CNOT order is correct.

  • Appendix A: Notation: The ˆZ operator has eigenvalues +1 and −1 for |g⟩ and |e⟩, and MZ returns these eigenvalues while projecting onto the corresponding eigenstates.The Hamiltonian is positively proportional to −ˆZ.
  • Appendix A: Notation: The notation defines ˆY as real and gives the modified commutation relations [ˆX, ˆY] = −2ˆZ, [ˆY, ˆZ] = −2ˆX, and [ˆZ, ˆX] = +2ˆY.The absence of i in ˆY interrupts the usual cyclic permutation of Pauli operators.
  • 8. The ˆS gate is another ˆZ-axis rotation, represented: The ˆT gate is a ˆZ-axis rotation, and is also called the π/8 gate.The appendix represents ˆT in the ˆZ basis and notes the π/8 formulation.
  • 8. The ˆS gate is another ˆZ-axis rotation, represented: A controlled-NOT applies ˆI to the target when the control is |g⟩ and ˆX to the target when the control is |e⟩.The first state is the control and the second is the target.
  • 8. The ˆS gate is another ˆZ-axis rotation, represented: A Toffoli gate applies ˆX to the target only when both controls are |e⟩; otherwise it applies ˆI.The Toffoli is a three-qubit controlled-controlled NOT gate.
  • 8. The ˆS gate is another ˆZ-axis rotation, represented: Each surface-code cycle measures a stabilizer formed by the product of the four neighboring physical-qubit ˆZ or ˆX operators.For example, ˆZsj = ˆZj,a ˆZj,b ˆZj,c ˆZj,d, with analogous construction for ˆX.
  • Appendix B: ˆZ and ˆ X stabilizer circuits: Appendix B simplifies stabilization to two data qubits, a and b, with one measure-Z and one measure-X qubit, requiring two CNOTs per measure qubit.The full four-qubit stabilization extends straightforwardly from this layout.
  • Appendix B: ˆZ and ˆ X stabilizer circuits: Given an arbitrary input, the circuit projects the data qubits onto one of four simultaneous eigenstates of ˆXa ˆXb and ˆZa ˆZb, with terminal measurements identifying the state.The circuit initializes the measure qubits, applies Hadamard and ordered CNOT operations, then performs terminal ˆZ measurements.

Appendix C: X-cut qubit initialization in a ˆZ eigenstate · Appendix D: Measuring an X-cut qubit in the ˆZL basis

Appendix C gives a stepwise procedure for initializing an X-cut qubit in |gL⟩, while Appendix D describes measuring an X-cut qubit in the ˆZL basis and restoring the array. Both procedures use temporary cuts, data-qubit ˆZ measurements, and stabilizer changes to preserve error tracking.

  • Appendix C: X-cut qubit initialization in a ˆZ eigenstate: Appendix C begins with an infinite lattice without qubit cuts, then turns off a column of four measure-X qubits to open a rectangular cut.The six adjacent measure-Z qubits are changed from four-terminal to three-terminal stabilizer measurements.
  • Appendix C: X-cut qubit initialization in a ˆZ eigenstate: Inside the cut, three data qubits are measured in the ˆZ basis to maintain error tracking, either directly or using idle measure-X qubits.The adjacent measure-Z qubits simultaneously use three-terminal stabilizer measurements.
  • Appendix C: X-cut qubit initialization in a ˆZ eigenstate: Resetting the data qubits inside the cut to |g⟩ initializes the logical qubit to |gL⟩.If step 2 is nondestructive, this reset can be omitted; the ˆZL eigenvalue is then the product of the measured ˆZ values.
  • Appendix C: X-cut qubit initialization in a ˆZ eigenstate: Appendix C completes initialization by restoring the two middle measure-X qubits and four-terminal measure-Z measurements while comparing successive outcomes for error detection.Because the internal data qubits were set to |g⟩, the reactivated measure-X qubits produce random outcomes.
  • Appendix D: Measuring an X-cut qubit in the ˆZL basis: Appendix D starts from an X-cut logical qubit in an arbitrary state, turns off the two intervening measure-X qubits, and changes neighboring measure-Z measurements to three-terminal form.The three data qubits inside the rectangular cut are then measured in the ˆZ basis.
  • Appendix D: Measuring an X-cut qubit in the ˆZL basis: The product ZaZbZc of the three individual data-qubit outcomes equals the logical ˆZL eigenvalue because ˆZL commutes with each individual ˆZj operator.Combining these measurements with three-terminal measure-Z outcomes preserves surface-code error tracking.
  • Appendix D: Measuring an X-cut qubit in the ˆZL basis: After measurement, the three data qubits are reset to |g⟩ when needed, and full four-terminal ˆZ stabilization plus all measure-X qubits are restored to destroy the logical qubit.The reset is unnecessary when the preceding ˆZ measurement has high fidelity for projecting onto ˆZ eigenstates; restored measure-X qubits report random outcomes.

Appendix E: Making a larger qubit · Appendix F: One-cell qubit move · Byproduct operators

The appendices describe how to construct a distance d = 8 logical qubit, move a Z-cut qubit by one cell through stabilizer changes and measurements, and track measurement-dependent sign changes with software-managed byproduct operators.

  • Appendix E: Making a larger qubit: A distance d = 8 logical qubit is made by turning off one stabilizer per qubit hole while defining logical operators from stabilizer products.The initial logical state is determined by the product Zs1Zs2Zs3Zs4 = ±1.
  • Appendix E: Making a larger qubit: The larger qubit is initialized by stopping measurements of four Z stabilizers inside the ZL loop, then turning off one interior X stabilizer and modifying neighboring X measurements.The four internal data qubits are projected onto a product of X eigenstates, either directly or using idle measure-Z qubits.
  • Appendix F: One-cell qubit move: A one-cell Z-cut qubit move extends ZL with the four single-qubit operators Z6Z7Z8Z9, producing an equivalent logical action up to the stabilizer outcome Z6789 = ±1.The extended loop encircles two cells during the move.
  • Appendix F: One-cell qubit move: The move turns off the Z6Z7Z8Z9 stabilizer, measures data qubit 6 in the X basis with outcome X6 = ±1, and compares neighboring measurements to maintain error detection.The neighboring X stabilizers are temporarily changed from four-terminal to three-terminal measurements.
  • Appendix F: One-cell qubit move: The move completes by turning on Z3Z4Z5Z6, restoring neighboring four-terminal X measurements, waiting a minimum of d cycles, and defining the translated logical operators.For larger-distance qubits, the intermediate procedure requires waiting d/4 rounded up surface code cycles before stabilization.
  • Byproduct operators: Measurement outcomes can reverse the signs of the translated logical operators: X6 = −1 affects X′L, while Z6789Z3456 affects Z′L.These sign changes arise from the measurements involved in extending and reestablishing the logical operators.
  • Byproduct operators: Byproduct operators encode these sign changes as logical bit- and phase-flips controlled by pX and pZ, rather than being physically applied to the logical qubit.The software control system instead changes the signs of subsequent logical measurements appropriately.

Appendix G: Multi-cell moves · Appendix H: Single qubit braid transformation

Appendix G specifies a multi-cell qubit shift through stabilizer extensions, measurements, reactivation, and byproduct corrections. Appendix H tracks the corresponding single-qubit braid transformations, showing that measurement-dependent signs are software-correctable and otherwise preserve the logical operators around stabilized regions.

  • Appendix G: Multi-cell moves: Appendix G: Multi-cell moves performs a multi-cell shift by extending the logical Z operator into an extended loop.The extension is denoted Ẑe_L and consists of a chain of Ẑ operators.
  • Appendix G: Multi-cell moves: The extended Z loop can differ in sign from the original logical operator according to pre-move stabilizer values.The sign depends on the product of the relevant pre-move stabilizer values.
  • Appendix G: Multi-cell moves: The shift disconnects internal data qubits by turning off enclosed Z stabilizers and converting bordering four-terminal X stabilizers to three-terminal stabilizers.Disconnected data qubits are measured along X, producing outcomes X1, X2 . . . Xn−1 equal to ±1 for comparison with prior stabilizer values.
  • Appendix G: Multi-cell moves: The new logical X and Z operators are defined from the measurement results and stabilizers, with the new Z loop algebraically equivalent to the nth Z stabilizer.The post-move Z stabilizers are reactivated and must be allowed at least d surface code cycles to establish stable values.
  • Appendix G: Multi-cell moves: Appendix G corrects move-induced signs by applying tracked byproduct operators to the ideal post-move logical state.The byproduct powers are defined as for the one-cell move and use pre- and post-move stabilizer information.
  • Appendix H: Single qubit braid transformation: The X-loop sign is determined by measurement products and equals the product of enclosed stable X stabilizers, because shared data-qubit operators cancel.The power pX records whether the transformed loop has the same or different sign from ẊL.
  • Appendix H: Single qubit braid transformation: The braid similarly transforms the Z loop through expansions and contractions, with sign changes tracked by pZ and corrected using software-tracked byproduct operators.Apart from these measurement-determined signs and byproducts, the braid around a fully stabilized array region leaves the two logical operators of the Z-cut unchanged.

Appendix I: Two-qubit braid transformation

The appendix demonstrates that braiding a Z-cut qubit through an X-cut qubit implements a logical CNOT, with the Z-cut qubit as control and the X-cut qubit as target. It derives the operator transformations, including braid reversal under repeated application up to byproduct operators.

  • Braid transformation: Braiding a Z-cut qubit through an X-cut qubit is equivalent to a logical CNOT, with the Z-cut qubit as control and the X-cut qubit as target.The same control-target assignment holds when braiding an X-cut qubit through a Z-cut qubit; braiding two qubits of the same cut type requires other logic.
  • X-operator transformations: Under the braid, ˆXL ⊗ˆXL transforms to ˆXL ⊗ˆIL because the two induced ˆXL operations on the X-cut qubit cancel.Repeating the same braid generally returns this operator pair to its original form, aside from byproduct operators.
  • X-operator transformations: The braid leaves ˆIL ⊗ˆXL unchanged, apart from byproduct operators, because the identity on the moving Z-cut qubit produces no operator trail.Dragging the hole around the closed loop therefore does not modify the X-cut qubit’s logical operator.
  • Z-operator transformations: The braid leaves ˆZL ⊗ˆIL unchanged on the logical operators, with only byproduct operators acquired by the wavefunction.The ˆZL loop does not enclose the X-cut qubit in a way that interacts with it.

Byproduct operators for two-qubit transformations … Appendix L: ˆSL and ˆTL distillation sub-circuits

The appendices specify how surface-code transformations track and correct braid-induced byproducts, execute a logical Hadamard, create and prepare short qubits, and analyze terminal S and T† distillation sub-circuits. These procedures rely on stabilizer manipulations, measurement outcomes, lattice deformation, and conditional operations.

  • Byproduct operators for two-qubit transformations: Braid-induced sign changes in logical Z operators are identified from stabilizer and data-qubit measurement outcomes and corrected with Z_L1 and Z_L2 byproducts.The correction powers satisfy p_Z1 = 1(0) and p_Z2 = 1(0) when the corresponding transformed operator does (not) acquire a sign change.
  • Appendix J: Logical Hadamard process: A logical Hadamard isolates the target qubit by turning off surrounding X stabilizers, reducing adjacent Z stabilizers, and measuring gap data qubits in Z.This preserves surface-code error tracking while preparing the logical qubit patch for subsequent deformation.
  • Appendix J: Logical Hadamard process: The Hadamard procedure deforms the Z_L loop across the isolated patch, creates a moat, and converts the geometry into a larger d = 7 array qubit with crossed logical operators.The resulting patch has Z_L crossing left-to-right and X_L crossing top-to-bottom, with measurements preserving error tracking.
  • Appendix J: Logical Hadamard process: Physical Hadamards swap X and Z eigenbases and logical-operator identities, after which two data–measure-qubit swaps realign the transformed patch with the surrounding array.The swaps occur between surface-code cycles, and the data qubits then hold the Hadamard-transformed logical state.
  • Appendix J: Logical Hadamard process: The Hadamard transformation is completed by recreating and moving the two Z-cut holes, waiting d surface-code cycles where required, and reunifying the patch with the array.The hole movement is split into two steps to avoid reducing the logical-qubit distance.
  • Appendix K: Short qubits: A short X-cut qubit is formed by disabling two X stabilizers around a single separating data qubit, measuring compatible Z-stabilizer products and that qubit in X, and preserving error tracking.The two holes are separated by qubit 5, while the product Z_s1 Z_s2 commutes with the simultaneous measurements.
  • Appendix K: Short qubits: The initialized short qubit is rotated about Z into a target logical state, with θ = π/2 for |Y_L⟩ and θ = π/4 for |A_L⟩, then enlarged and purified for S_L or T_L use.The two holes are separated and enlarged after stabilization restarts, improving protection before distillation.
  • Appendix L: ˆSL and ˆTL distillation sub-circuits: Appendix L analyzes logical S and T† distillation terminal sub-circuits followed by M_X, using an expanded circuit with conditional measurements for a parallel treatment.The Fig. 30-style |Y_L⟩ circuit implements S_L only about half the time, whereas the Fig. 29 S_L circuit is deterministic.

ˆS circuit displaying the logical CNOT and measurements M ′ · Appendix M: Estimating the time and size of a factoring circuit

The ˆS and ˆT † circuits use conditional corrections and measurement-sign changes to implement or verify non-Clifford operations, with X and Z ancilla errors directly detected. For factoring a 2000-bit number, the surface-code design requires 26.7 hours, roughly 2 × 10^12 distilled |AL⟩ states, and about 800,000 physical qubits for state generation.

  • ˆS circuit displaying the logical CNOT and measurements M ′: The desired M_X measurement equals the product of the two sub-measurements, including a minus sign when the circuit produces ˆX ˆZ ˆS instead of ˆS.This follows from M_X[ˆX|ψ⟩] = M_X[|ψ⟩] and M_X[ˆZ|ψ⟩] = −M_X[|ψ⟩].
  • ˆS circuit displaying the logical CNOT and measurements M ′: ˆX and ˆZ ancilla errors reverse the final M ′X measurement sign, while ˆY errors are undetectable but do not affect distillation.The circuit therefore directly detects all distillation-relevant errors through the measurement sign.
  • ˆS circuit displaying the logical CNOT and measurements M ′: For the ˆT † circuit, incorrect |A⟩-ancilla outputs are signalled by a sign change in M ′X, with one outcome accepted and the other discarded.The circuit yields the desired ˆT † output with ˆX on the measurement about half the time and −M ′X the other half.
  • Appendix M: Estimating the time and size of a factoring circuit: For a 2000-bit factorization, modular exponentiation uses 40N^3 sequential Toffoli gates and requires 120N^3 tM total time.Each Toffoli uses seven ˆTL gates, arranged as three parallel gates, one gate, then three parallel gates.
  • Appendix M: Estimating the time and size of a factoring circuit: 26.7 hours are required to factor a N = 2000 bit number when tM = 100 ns.The estimate assumes each ˆTL gate completes in one measurement time and each Toffoli therefore takes 3tM.
  • Appendix M: Estimating the time and size of a factoring circuit: 2 × 10^12 |AL⟩ states are consumed, and their final error rate should be much less than PA ≈4 × 10−13.The consumption is 280N^3 states for N = 2000.
  • Appendix M: Estimating the time and size of a factoring circuit: Two distillation stages reduce a 0.5% injection error to p1 = 35p3I ≈4 × 10−6 and p2 = 35p3_1 ≈3 × 10−15, below PA.The estimates assume error-free distillation circuits for the first two stages.
  • Appendix M: Estimating the time and size of a factoring circuit: About 800,000 physical qubits and 500 surface code cycles are required to generate one sufficiently purified |AL⟩ state, while an AA factory produces two states every approximately 100 µs.The footprint and timing derive from reusing the first-stage footprint for the second stage and a 200 ns surface code cycle.
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