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Explicit construction of low-overhead gadgets for gates on quantum LDPC codes
Paul Webster, Samuel C. Smith, Lawrence Z. Cohen
TL;DR
QLDPC codes could reduce quantum-computing overhead, but efficient arbitrary logical operations remain challenging. The paper gives fixed gadgets for measuring arbitrary logical Pauli operators and applies them to generalised bicycle codes. For distances 10 ≤ d ≤ 24, the construction reduces space overhead by at least an order of magnitude versus corresponding surface-code architectures without increasing time overhead.
Problem
Implementing universal gates on QLDPC codes requires efficient methods to address particular logical qubits and measure arbitrary logical Pauli operators without impractical dynamic hardware reconfiguration.
Source: §I · p. 1Method
The paper constructs fixed gadgets for seed operators and uses code automorphisms, relocation, and bridging to measure arbitrary logical Pauli operators.
Source: §I · p. 1, B. Measurement of Arbitrary Logical… · p. 2, §2 · p. 3Results
For generalised bicycle codes with 10 ≤ d ≤ 24, the construction reduces space overhead by a factor of 11–24 relative to same-distance surface-code architectures without increasing time overhead.
Source: §I · p. 1Takeaways & Limitations
The construction offers broadly applicable low-overhead logical-Pauli measurement on QLDPC codes when code automorphisms provide a small complete seed set.
Source: §I · p. 1, §IV · p. 4
Abstract
from arXiv · showhide
Quantum low-density parity check (QLDPC) codes can significantly reduce the overhead of quantum computing, provided the methods for performing logical operations do not require substantial space and time resources. A popular method for performing logical operations is by measuring logical Pauli operators. We present a simple, explicit construction for fixed gadgets that can measure arbitrary logical Pauli operators on QLDPC codes when dynamically connected to the code block. We apply this construction to a family of generalised bicycle codes with distances relevant to utility-scale quantum computation ($10\leq d \leq 24$) and show that it reduces the space overhead by at least an order of magnitude compared to corresponding surface code architectures, without increasing the time overhead.
I. INTRODUCTION
QLDPC codes may reduce the overhead of fault-tolerant quantum computing, but implementing arbitrary logical operations efficiently remains challenging. The paper addresses this by constructing fixed, low-overhead gadgets that measure arbitrary logical Pauli operators.
I. INTRODUCTION
- QLDPC codes can encode many logical qubits per code block through nonlocal connections, potentially reducing the hundreds-to-thousands of physical qubits otherwise used per logical qubit.Surface-code architectures may require a million or more qubits for applications such as quantum chemistry or cryptography [2] [3] [4].Source: §I · p. 1
- Universal gates are difficult on QLDPC codes because particular logical qubits within a code block must be addressed, motivating ancillary gadgets for logical-Pauli measurements.Together with magic states, such measurements suffice for universal quantum computation.Source: §I · p. 1
- Dynamically rearranging bespoke ancilla gadgets may be impractical because the required hardware changes depend on preceding measurement outcomes and cannot be pre-programmed.This motivates fixed rather than dynamically reconfigured gadgets.Source: §I · p. 1
- Existing approaches leave open how to construct a low-overhead gadget that efficiently measures all logical Pauli operators on a given QLDPC code [15].Some code-specific gadgets measure only subsets of logical operators, creating time overhead from compilation [10] [17].Source: §I · p. 1
- The construction reduces overhead by a factor of 11–24 relative to same-distance surface-code architectures for distances 10 ≤ d ≤ 24.The comparison targets distances considered relevant to early utility-scale quantum computation.Source: §I · p. 1
- The paper relocates and bridges fixed gadgets for a constant number of seed operators, giving total gadget size ˜O(d) for suitable code families instead of ˜O(n) for a general extractor system [15].The construction is applied to generalised bicycle codes with the stated 11–24 overhead reduction.Source: §I · p. 1
II. CONSTRUCTION
The construction proceeds in two stages: first, a gadget is built for one logical Pauli operator, then fixed gadgets are combined to measure arbitrary logical Pauli operators.
II. CONSTRUCTION
- The section constructs a low-overhead gadget for arbitrary logical Pauli measurements on a QLDPC code.Section II A handles one logical operator, while Section II B extends the construction to arbitrary operators.Source: §II · p. 1
A. Gadget for Measuring a Particular Logical Operator
A gadget for one logical operator is built from code checks and support qubits, with added checks fixing gauge freedom; the product of gadget-check outcomes gives the logical measurement.
A. Gadget for Measuring a Particular Logical Operator
- For each relevant Z check S_j, the gadget adds a qubit κ_j to that check’s support.These gadget qubits connect the original checks to the added X-type checks.Source: A. Gadget for Measuring a Particular… · p. 2
- For each physical qubit q_i in the logical operator’s support, the gadget adds an X-type check χ_i connected to q_i and the κ_j associated with checks containing q_i.The construction assumes an X operator on the support; arbitrary Pauli operators are handled using single-qubit Clifford gates.Source: A. Gadget for Measuring a Particular… · pp. 1–2
- Additional Z-type gadget checks are added from a basis of the kernel to fix the gauge degrees of freedom introduced by the gadget.Steps 1 and 2 form the dual of the restricted code Tanner graph, while Step 3 removes the resulting extra gauge freedom.Source: A. Gadget for Measuring a Particular… · p. 2
- The logical X measurement outcome equals the product of the corresponding gadget-check outcomes χ_i.The construction uses the commutation relation between the logical operator and the relevant Z checks.Source: A. Gadget for Measuring a Particular… · p. 2
- The combined code-gadget stabilisers commute because the relevant supports intersect evenly, while the remaining checks act trivially or commute by construction.This establishes that the combined system is a valid stabiliser code.Source: A. Gadget for Measuring a Particular… · p. 2
- The gadget remains distance preserving when its X Tanner graph has Cheeger constant h ≥ 1; extra qubits and checks can increase expansion when necessary [10].For the paper’s gadgets, the relevant condition reduces to the standard graph Cheeger constant, and O(n) edges can increase it.Source: A. Gadget for Measuring a Particular… · p. 2
B. Measurement of Arbitrary Logical Operators
Code automorphisms provide the mechanism for reusing a gadget construction across logical operators, while bridging combines gadgets to measure products of operators.
B. Measurement of Arbitrary Logical Operators
- A stabiliser code automorphism permutes physical qubits while commuting with the stabiliser group, enabling the particular-operator gadget method to target arbitrary logical operators.This supplies the fixed-gadget reuse mechanism for different logical measurements.Source: B. Measurement of Arbitrary Logical… · p. 2
1. Measuring Many Logical Operators With One Gadget
A seed operator’s gadget can be relocated through code automorphisms to measure every logical operator in its orbit. A complete seed set, combined with gadget bridging, therefore suffices to measure any logical Pauli operator.
1. Measuring Many Logical Operators With One Gadget
- A logical orbit is the set of logical actions generated from an operator by a group of code automorphisms.Source: §1 · p. 2
- A single gadget for a seed operator measures any logical operator in its orbit when connected to the corresponding permuted code qubits and checks.Code automorphisms preserve the Tanner graph, so relocating the gadget preserves its measurement function.Source: §1 · pp. 2–3
- A complete seed set is a collection whose logical orbits combine to generate the full Pauli group on the logical space.One gadget per seed, together with bridging, can then measure any logical operator.Source: §1 · p. 3
2. Bridging Gadgets to Measure Arbitrary Logical Operators
Bridging combines gadgets that measure separate logical operators into a gadget for their product, while preserving commutation and distance under the stated construction conditions.
2. Bridging Gadgets to Measure Arbitrary Logical Operators
- For disjoint-support operators, bridging uses w additional data qubits and w −1 checks, and iteratively measures m operators with O(mw) added qubits.The construction uses a common bridge; SkipTree controls check weights and qubit degrees, while distance is preserved when h ≥1.Source: §2 · p. 3
- Overlapping supports are handled by merging same-type checks in pairs and then merging any remaining X- and Z-type checks.The resulting checks commute, and the merge preserves distance because it does not decrease the bridged graph’s Cheeger constant.Source: §2 · p. 3
III. EXAMPLE APPLICATION
The construction is applied to generalised bicycle codes, whose cyclic-shift automorphisms generate large logical orbits and enable a small complete seed set. These codes have substantially higher rates than same-distance rotated surface codes over the stated distance range.
III. EXAMPLE APPLICATION
- Generalised bicycle codes are specified by a lift l and sets A,B ⊆ Z_l, with two sectors of l physical qubits for n = 2l total qubits.Source: §III · p. 3
- Simultaneous cyclic shifts form a Z_l automorphism group, allowing each seed operator to have a logical orbit of up to l elements.The shifts preserve the stabiliser group because they map each generator to another translated generator.Source: §III · p. 3
- For 10 ≤d ≤60, the generalised bicycle codes have rates approximately ten to twenty times greater than rotated surface codes of the same distance.The family has l = 2r −1 and weight-six check operators, with conjectured parameters ⟨2(2r −1), 2r, r +(r −4)^2⟩.Source: §III · pp. 3–4, Figure 1 · p. 4
- The logical space splits into two subspaces of l X-type and l Z-type operators, enabling a complete seed set of four operators.The construction explicitly achieves this using two logical X-type and two logical Z-type seed operators.Source: §III · p. 4
IV. CONCLUSION
The paper presents an explicit, broadly applicable construction for low-overhead QLDPC measurement gadgets. Its overhead is low when code automorphisms provide a small complete seed set, with possible extensions beyond the demonstrated code family.
IV. CONCLUSION
- The construction offers order-of-magnitude overhead reductions relative to surface-code architectures when applied to the studied generalised bicycle codes.Source: §IV · p. 4
- The method is not specific to generalised bicycle codes and applies wherever QLDPC code automorphisms allow a small complete seed set.The conclusion identifies bivariate bicycle, two-block group algebra, and lifted product codes as possible extensions.Source: §IV · p. 4
Appendix A: Seed operators
The appendix lists four generalized bicycle code instances and their seed operators, with logical orbits covering all logical Pauli operators on designated halves of the logical qubits.
Appendix A: Seed operators
- For each code, the logical orbit of X̄1 and Z̄1 contains all logical Pauli operators on the first k/2 logical qubits.Source: Appendix A · p. 6
- The logical orbit of X̄k/2+1 and Z̄k/2+1 contains all logical Pauli operators on the last k/2 logical qubits.Source: Appendix A · p. 6
- The four listed code instances use l values 31, 63, 127, and 255, with specified A and B sets for each case.Source: Appendix A · p. 6
- For the l = 31 instance, explicit support sets are given for the seed operators X̄1, Z̄1, X̄6, and Z̄6.Source: Appendix A · p. 6