PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum Measurements and Gates by Code Deformation

H. Bombin, M. A. Martin-Delgado

arXiv:0704.2540v2quant-phcond-mat.str-el

TL;DR

Fault-tolerant computation is commonly organized around code blocks, transversal operations, and destructive ancillary measurements, while surface codes restrict transversal gates. The paper uses code deformation to perform operations in a single-layer surface code; it demonstrates gates, initialization, and measurements while keeping interactions local in two dimensions.

  • Problem

    The usual fault-tolerant setup uses encoded code blocks, transversal operations, and destructive measurements of ancillary blocks, while surface codes do not support other gates transversally.

    Source: §I · pp. 2–3

  • Method

    The paper uses code deformation to create, manipulate, and measure encoded qubits in a single-layer surface code.

    Source: §I · p. 3, §IV · p. 10

  • Results

    The demonstrated operations include initialization in |+⟩ or |1⟩, Z- or X-basis measurements, CNot gates, and Hadamard gates without exposing encoded qubits to errors.

    Source: §V · p. 18

  • Takeaways & Limitations

    The operations can be used to distill noisy states obtained through progressive growth, including magic states for universal quantum computation.

    Source: §V · p. 18

Abstract

from arXiv · show

The usual scenario in fault tolerant quantum computation involves certain amount of qubits encoded in each code block, transversal operations between them and destructive measurements of ancillary code blocks. We introduce a new approach in which a single code layer is used for the entire computation, in particular a surface code. Qubits can be created, manipulated and non-destructively measured by code deformations that amount to `cut and paste' operations in the surface. All the interactions between qubits remain purely local in a two-dimensional setting.

I. INTRODUCTION

Fault-tolerant proposals can require non-local physical gates, and dimensionality matters when devices support only local interactions. The paper uses code deformation to obtain non-destructive measurements and CNot gates from a single-layer surface code in a 2D setting.

I. INTRODUCTION

  • Code deformation changes a code through successive stabilizer changes, enabling gates and changes in the number of encoded qubits for initialization and measurement.The paper says these ideas are natural for topological codes and could also apply to other error-correcting codes.Source: §I · p. 2
  • Non-local physical gates are a practical problem for fault-tolerant proposals, and locality differs across 1D, 2D, and 3D settings.The paper notes that local elementary gates in some devices constrain the gates available between physical qubits.Source: §I · p. 2
  • Surface codes are local in 2D, but their standard layered approach implements CNot transversally and no other gate transversally.Color codes offer a 2D alternative with transversal Clifford gates; 3D extensions can have transversal properties for universal computation.Source: §I · pp. 2–3
  • A single-layer surface code can support non-destructive measurements and CNot gates through deformation, addressing the limits of the standard surface-code approach.The paper also presents this as a route to non-trivial gates in a 2D setup.Source: §I · pp. 2–3
  • The paper distinguishes its approach from cluster-state work by using 2D topological-code properties without additional structure and considering varied surface topologies.It also says the deformation approach fits the error-correction scheme in and can be applied to other codes, including those in,.Source: §I · p. 3
  • The approach assumes active quantum error correction through external operations, rather than a Kitaev Hamiltonian associated with the code.The text contrasts this with proposals that rely on hardware-level topological protection.Source: §I · p. 4

II. CODE DEFORMATION

Code deformation uses small, progressive changes from one stabilizer code to another to initialize, transform, or measure encoded qubits. For surface codes, these changes can be geometrical and can be implemented through stabilizer measurements under stated error-correction conditions.

II. CODE DEFORMATION

  • Code deformation consists of small, progressive changes from one stabilizer code to another, intended to initialize, transform, and measure encoded qubits.The paper introduces it as an operation on stabilizer codes.Source: §II · p. 4

A. Stabilizer codes

  • A stabilizer code encodes k qubits when its stabilizer has n − k independent generators, and its distance measures the minimum weight of a normalizer element outside the stabilizer.The passage relates larger distance to greater noise resilience.Source: A. Stabilizer codes · p. 5

B. Code transformations

  • When only a small number of stabilizer generators change, the transformation is a code deformation; in surface codes, localized shape changes that preserve topology are called smooth deformations.The encoded Pauli evolution determines the encoded unitary, and successive partial transformations can implement Clifford gates.Source: B. Code transformations · p. 6
  • A code can be enlarged with extra fixed-state qubits by adding a stabilizer generator for each, allowing the transformation description to cover codes with different physical-qubit counts.The passage gives a Z stabilizer on each added qubit as one example.Source: B. Code transformations · p. 6
  • Earlier surface-code deformation work used explicit local CNot gates, while the alternative described here measures new stabilizers and accounts for potentially undesired measurement outcomes.When error correction monitors errors over repeated rounds, the text says correction of those outcomes may be unnecessary.Source: B. Code transformations · p. 6
  • Stabilizer-measurement deformations fit repeated error-correction rounds when errors can be corrected with high probability and measurements remain local relative to encoded Pauli supports.The text says a suitable unitary must be applied directly if errors cannot be corrected successfully with high probability.Source: B. Code transformations · p. 7

C. Initialization and measurement

Non-smooth deformations change the number of encoded qubits by changing surface-code topology. Increasing that number initializes a qubit in a definite state, while decreasing it measures an encoded operator.

C. Initialization and measurement

  • Unlike smooth deformations, these operations change the number of encoded qubits through changes in surface topology.They correspond to increasing or decreasing the number of nontrivial cycles.Source: C. Initialization and measurement · pp. 7–8
  • Adding an encoded qubit uses a deformation that adds a stabilizer generator and fixes a nontrivial encoded Pauli operator T to eigenvalue one.This initializes the new qubit in a definite way.Source: C. Initialization and measurement · p. 7
  • Removing an encoded qubit maps a nontrivial encoded operator T into the new stabilizer, thereby measuring T.In surface codes, these deformations decrease the number of nontrivial cycles.Source: C. Initialization and measurement · pp. 7–8
  • Errors in the operator being initialized or measured do not harm the intended operation, because initialization fixes its eigenvalue and measurement removes coherence between its outcomes.Topology changes can expose some degrees of freedom to errors, but the relevant errors correspond to the operator being initialized or measured.Source: C. Initialization and measurement · p. 8

III. SURFACE CODES

Surface codes place physical qubits on a two-colored lattice and define the stabilizer with plaquette operators. Encoded Pauli operators can be represented by strings whose equivalence and crossings determine their action on encoded states.

III. SURFACE CODES

  • Physical qubits occupy lattice sites, with X plaquette operators on dark plaquettes and Z plaquette operators on light plaquettes generating the stabilizer.The plaquette operators commute because of the lattice properties.Source: §III · pp. 8–9
  • Dark strings represent Z operators and light strings represent X operators; strings ending at matching-color borders count as closed.These strings commute with the stabilizer and represent normalizer operators.Source: §III · p. 9
  • Boundary strings generate stabilizers, while non-boundary closed strings represent encoded Pauli operators modulo stabilizers.Same-color strings differing by a boundary are equivalent, and opposite-color strings anticommute when they cross an odd number of times.Source: §III · pp. 9–10
  • A rectangular surface code encodes one qubit using nontrivial dark and light strings as its Z and X operators.Increasing the rectangle's width and height makes the code more resilient to Z and X errors, respectively.Source: §III · p. 10

IV. SURFACE CODE DEFORMATION

Surface-code deformation reshapes the code by changing its boundaries and stabilizers while leaving the underlying physical-qubit lattice fixed. Elementary changes are implemented by measuring the new plaquette operators and interpreting outcomes through error correction.

IV. SURFACE CODE DEFORMATION

  • Deformation changes the surface's dark and light borders and the stabilizers defining the code, not the fixed underlying physical-qubit lattice.Borders can be moved, glued together, or created by cutting the surface.Source: §IV · p. 10
  • Erasing a qubit at a light border removes a light plaquette operator and changes two dark plaquette operators.The new two-site plaquette operators are measured; if their eigenvalues are negative in the absence of errors, Z corrections are applied to marked qubits.Source: §IV · pp. 10–11
  • When errors may occur during deformation, the new plaquette measurements are interpreted at the error-correction stage.The inverse deformation requires corrections on the marked qubit.Source: §IV · p. 11

A. Smooth deformations

Smooth deformations preserve surface topology and the number of encoded qubits while inducing gates through the motion of encoded string operators. Moving a light hole around a dark one implements a source-controlled CNot, although deformations cannot implement every Clifford operation.

A. Smooth deformations

  • A smooth deformation preserves topology and cannot change the number of encoded qubits, but can implement a unitary gate.The deformation maps the surface back to its original form.Source: A. Smooth deformations · p. 11
  • Moving a light hole around a dark one gives a CNot with the first qubit as source.The encoded Pauli operators transform as UZ1U† ∼S Z1Z2 and UX2U† ∼S X1X2, while X1 and Z2 are unchanged.Source: Figure 3 · p. 12, A. Smooth deformations · p. 12
  • The deformation acts on encoded Pauli operators by mapping their representative strings through the continuous surface map f.The gate action follows from how those string operators change under the deformation.Source: A. Smooth deformations · p. 12
  • Deformations cannot implement Hadamard gates because they do not mix X and Z operators.The allowed string evolution is more restrictive than general Clifford operations.Source: A. Smooth deformations · p. 12

B. Non-smooth deformations

Code deformations can measure encoded operators by changing a surface code’s topology, with cuts mapping operator eigenvalues to stabilizer violations. The deformation can be reversed, leaving the encoded state projected.

B. Non-smooth deformations

  • Reversing a drastic deformation leaves the encoded state projected, implementing a non-destructive measurement of an encoded operator.The surface can be temporarily contracted to expose an encoded operator while the measurement remains protected by the code.Source: B. Non-smooth deformations · p. 13
  • Removing a qubit that connects two surface pieces changes the topology and can measure an encoded Pauli operator.In the example, removing q and adding stabilizer Xq maps the encoded X1 measurement to the new code’s stabilizer conditions.Source: B. Non-smooth deformations · p. 13
  • Cutting along a light string between two light borders measures Xs; cutting along a dark string measures Zs.A local cut maps the measured string’s eigenvalue to the parity of stabilizer violations appearing on either side.Source: B. Non-smooth deformations · pp. 13–14
  • For the illustrated cut, the lower diagrams show the X = −1 outcome; read bottom-up, discontinuous deformations introduce a qubit initialized with X = 1.The figure caption also identifies border contraction and hole contraction as examples.Source: B. Non-smooth deformations · p. 14

C. Error correction

Surface-code error correction infers error worldlines from repeated stabilizer measurements and succeeds with high probability below threshold when inferred and actual worldlines are homologically equivalent. Deformations leave this procedure basically unchanged, with corrections accounted for when recovering measurement values.

C. Error correction

  • Below threshold, error correction can infer worldlines with high probability, provided actual and inferred worldlines are homologically equivalent.Repeated stabilizer measurements produce time slices for X- and Z-type violations, arranged as (2+1)-D worldlines.Source: C. Error correction · p. 15
  • During a deformation that measures and loses an encoded qubit, recover the measurement value only after accounting for errors and checking relevant stabilizer violations.The error-correction procedure otherwise remains basically unchanged during deformations.Source: C. Error correction · p. 15

D. A particular implementation

The implementation associates encoded qubits with holes in a surface code; the passage also describes an initialization procedure for |1⟩.

D. A particular implementation

1. Encoded qubits and initialization

Encoded qubits are represented by holes with a specified border structure, and initialization grows paired borders after puncturing the code. A CNot gate is obtained by pasting borders, winding a source hole around a target hole, and cutting to restore the qubits’ original shapes.

1. Encoded qubits and initialization

  • Encoded qubits are holes with dark-light-dark-light borders, and a suitable border can be shortened to measure encoded Z or X.To initialize |1⟩ or |+⟩, puncture the code to create a dark or light hole, then grow a pair of light or dark borders along its boundary.Source: §1 · p. 16

2. CNot gate

  • The CNot procedure uses paired paste operations to form two-holed qubits before a hole is moved around a target hole.The figure sequence depicts the qubits before the gate, the pasting and hole movement, and the final cuts.Source: §2 · p. 17
  • A source hole winds around a target hole between paste and cut operations, and the encoded operators then evolve as for a CNOT.The initial pasting creates two additional encoded qubits; cutting to restore the original shapes measures them.Source: §2 · pp. 16–17

3. Qubit disconnection and reconnection

The section describes disconnecting a qubit from the surface code by a chosen deformation and reconnecting it by reversing the procedure. It then states that code deformation supports gates, initialization, measurements, and state distillation in surface codes.

3. Qubit disconnection and reconnection

  • The chosen cut disconnects a qubit, and reversing the procedure reconnects it; the cut string operator has eigenvalue 1.After the cut, the isolated qubit occupies an inner lattice equivalent up to deformation to the lattice in Fig. 2(a).Source: §3 · pp. 17–18, Figure 7 · p. 18
  • The isolated qubit's lattice matches the one used for transversal initialization, destructive X and Z measurements, and encoded Hadamard gates.The match is up to smooth deformations.Source: §3 · pp. 17–18

V. CONCLUSIONS

  • Code deformation supports initialization, X- or Z-basis measurements, CNot and Hadamard gates without exposing encoded qubits to errors.These operations can also distill noisy states obtained through progressive growth, including magic states for universal quantum computation.Source: §V · p. 18
Loading 0704.2540v2…