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Universal Quantum Computation and Leakage Reduction in the 3-Qubit Decoherence Free Subsystem
Bryan H. Fong, Stephen M. Wandzura
TL;DR
The 3-qubit DFS needs explicit exchange-only gates and a way to reduce leakage errors. The paper gives exact exchange pulse sequences for a full subsystem CNOT and leakage reduction, requiring 22 gates in 13 time steps and 30 gates in 20 time steps, respectively.
Problem
Local decoherence can produce leakage from the 3-qubit DFS, which must be reduced to encoded errors during quantum error correction.
Source: §1 · p. 2Method
The paper derives the required operator forms in the total angular momentum basis and searches for exchange-only pulse sequences for the CNOT and leakage-reduction operators.
Source: §2 · p. 5, §3 · p. 9, §4 · p. 14Results
22 exchange gates in 13 time steps implement the full subsystem CNOT, and the leakage-reduction sequence uses 30 gates in 20 time steps.
Source: §1 · pp. 2–3Takeaways & Limitations
The sequences provide two required operations for 3-qubit DFS computation; the leakage-reduction sequence is the first explicit exchange-only sequence of its kind.
Source: §1 · p. 3, §6 · p. 16
Abstract
from arXiv · showhide
We describe exchange-only universal quantum computation and leakage reduction in the 3-qubit decoherence free subsystem (DFS). We discuss the angular momentum structure of the DFS, the proper forms for the DFS CNOT and leakage reduction operators in the total angular momentum basis, and new exchange-only pulse sequences for the CNOT and leakage reduction operators. Our new DFS CNOT sequence requires 22 pulses in 13 time steps. The DFS leakage reduction sequence, the first explicit leakage reduction sequence of its kind, requires 30 pulses in 20 time steps. Although the search for sequences was performed numerically using a genetic algorithm, the solutions presented here are exact, with closed-form expressions.
1 Introduction
The paper addresses the need for explicit exchange-only gates and leakage recovery in the 3-qubit decoherence-free subsystem (DFS). It presents shorter full-subsystem CNOT and leakage-reduction pulse sequences.
1 Introduction
- Exchange-only encoding can avoid slow, technically challenging single-electron operations, and prior work established universality using exchange or other two-body interactions.Experiments on a single three-spin exchange-only encoded qubit had also been reported.Source: §1 · pp. 1–2
- Earlier explicit sequences included a locally equivalent CNOT for the 3-qubit subspace, whereas the subsystem requires a full CNOT sequence.The earlier 19-pulse sequence was not valid for the entire subsystem.Source: §1 · p. 2
- For the full encoded 3-qubit DF subsystem, the analytic CNOT sequence uses 22 exchange gates in 13 time steps; the first exchange-only leakage-reduction sequence uses 30 gates in 20 time steps.Source: §1 · pp. 2–3
- Local decoherence can cause encoded errors and leakage, and leakage must be reduced to encoded errors during quantum error correction.The subsystem protects against global decoherence, but not local mechanisms such as nuclear hyperfine and electron-electron dipole-dipole coupling.Source: §1 · p. 2
2 Angular Momentum Structure of DFS
The 3-qubit DFS is described using total spin, the first-two-qubit spin, and total spin-z. These quantum numbers distinguish leakage, logical information, and gauge, and exchange preserves some while acting on the encoded state.
2 Angular Momentum Structure of DFS
- The total spin S distinguishes valid unleaked states from leaked states, while S1,2 gives logical information and Sz gives the gauge state.Valid subsystem states factor into encoded and gauge parts; global decoherence changes the gauge state rather than encoded information.Source: §2 · p. 4
- Exchange commutes with total spin and total spin-z, but changes S1,2 and therefore can act on the encoded state.Exchange between qubits 1 and 2 generates an encoded z rotation; exchange between qubits 2 and 3 generates a rotation about the stated axis.Source: §2 · p. 5
- For two DFS qubits, the total angular momentum basis uses Stot, Sz,tot, each qubit's total spin, and each qubit's first-two-qubit spin.Conservation of Stot and Sz,tot yields a partially diagonalized, block-structured exchange operator.Source: §2 · pp. 5–6
- The six-qubit Hilbert space contains five spin-0, nine spin-1, five spin-2, and one spin-3 subspaces.Exchange-constructed operators do not mix different Stot or Sz,tot subspaces; blocks with the same Stot and different Sz,tot have identical matrix entries.Source: §2 · pp. 5–6
3 DFS CNOT Pulse Sequence
The exchange-pulse sequence implements a full CNOT on the 3-qubit DFS subsystem, acting on logical information independently of gauge state. Its matrix constraints preserve the encoded operation while preventing coupling between leaked and unleaked states.
3 DFS CNOT Pulse Sequence
- The CNOT must act on the logical information in both DFS qubits, independently of their gauge state.The encoded information is carried by the quantum numbers S_A,1,2 and S_B,1,2.Source: §3 · p. 6
- In the spin-0 and spin-1 blocks, the encoded states carry the CNOT while leaked states remain uncoupled from unleaked states.The spin-0 block has a 4 × 4 encoded CNOT block; the spin-1 blocks also require an encoded CNOT block and do not couple leaked and unleaked states.Source: §3 · p. 8
- The exchange-gate search uses an objective function whose zero value requires specified encoded matrix entries to have unit modulus and a common phase.The product matrix is ordered using the spin-0 basis vectors 1–5 and spin-1 basis vectors 6–14.Source: §3 · p. 8
- 22 pulses in 13 time steps give an analytic, full (not merely locally equivalent) CNOT for the 3-qubit DF subsystem.The spin-0 and spin-1 blocks share a global phase and satisfy the stated CNOT constraints.Source: §3 · pp. 9–10, Figure 1 · p. 9
- Removing four gates gives an 18-pulse, 11-step locally equivalent CNOT, shorter than the locally equivalent CNOT sequence for the DF subspace alone.Only nearest-neighbor exchange gates are permitted.Source: §3 · pp. 9–10
4 DFS Leakage Reduction Pulse Sequence
The paper specifies how exchange operations must preserve or repair the encoded state of a potentially leaked DFS qubit, then presents an exact leakage-reduction sequence. The sequence uses 30 pulses in 20 time steps, and its objective function tests whether the resulting blocks satisfy the required constraints.
4 DFS Leakage Reduction Pulse Sequence
- If A is unleaked, leakage reduction preserves its encoded quantum number but may alter its gauge; if leaked, A returns to the unleaked subspace.The fiducial qubit B must be unleaked, while its final state is unconstrained and may be leaked.Source: §4 · pp. 10, 12
- For an unleaked DFS qubit B, the spin-0 matrix and spin-1 columns 1–4 constraints together ensure the identity on DFS qubit A’s encoded quantum number regardless of gauge state.In spin 1, columns 3 and 4 match columns 1 and 2 with shifted entries; in spin 0, the fifth basis vector is decoupled, ensuring no leakage from A.Source: §4 · p. 11
- In spin 2, columns 3 and 4 map leaked A states to states where A is unleaked, provided B is unleaked.The spin-3 action is constrained only by unitarity.Source: §4 · pp. 11–12
- The search imposes matching f and h matrices, a nonrequired constraint that makes leaked A reset to a factorizable encoded-and-gauge state and enabled an analytic solution.The objective is zero when the specified blocks satisfy their unitarity and matching constraints.Source: §4 · pp. 12, 14
- 30 pulses in 20 time steps implement the leakage-reduction operator in an exact sequence with closed-form exchange-gate powers.The sequence satisfies the operator forms in the total angular momentum basis.Source: Figure 2 · p. 13, §4 · p. 14
5 Genetic Algorithm
The genetic search evolves exchange-gate configurations toward low objective values while penalizing gate count. Mutations alter or refine configurations, mating combines parents, and weighted selection favors low-objective, shorter sequences.
5 Genetic Algorithm
- The genetic algorithm minimizes objective functions for the desired gate forms while penalizing sequence length to trade off accuracy and gate count.The search uses random mutations and mating followed by selection; the computationally intensive search can be parallelized.Source: §5 · pp. 14–15
5.1 Mutations
- Five mutation types are chosen at random, including inserting or deleting gates and refining swap powers.Refinement can target one gate, an adjacent pair, or all powers in a configuration.Source: §5.1 · pp. 15–16
5.2 Mating
- Mating forms offspring by joining the beginning of one parent configuration to the end of another, then applying RefineAll.Without mating, the population’s diversity suffers.Source: §5.2 · p. 16
5.3 Natural Selection
- Selection weights configurations by the inverse of the objective plus gate penalty, favoring low-objective, low-gate-count sequences.After selection, adjacent gates on the same qubit pair with no intervening noncommuting gates are combined by adding their powers.Source: §5.3 · p. 16
6 Conclusion
The paper reports exact exchange-only pulse sequences for encoded CNOT and leakage reduction in the 3-qubit DFS. It describes the CNOT as the most efficient sequence yet found for the subspace or subsystem, and the leakage-reduction sequence as the first explicit one for an exchange-only encoding.
6 Conclusion
- The CNOT sequence is the most efficient yet found for the 3-qubit subspace or subsystem, while the leakage-reduction sequence is the first explicit sequence for an exchange-only encoding.Other solutions with additional constraints on leaked-state evolution may have better error-propagation properties with fault-tolerant error correction.Source: §6 · p. 16