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Dynamically protected cat-qubits: a new paradigm for universal quantum computation
Mazyar Mirrahimi, Zaki Leghtas, Victor V. Albert, Steven Touzard, Robert J. Schoelkopf, Liang Jiang, Michel H. Devoret
TL;DR
The paper addresses how to build protected logical qubits and universal gates without repeatedly decoding information onto vulnerable physical qubits. It uses multi-photon driven dissipation to stabilize cat-state encodings, supplies Hamiltonian gate operations, and proposes superconducting-circuit implementations; four-component cats with continuous parity measurements additionally track single-photon loss.
Problem
Protected oscillator memories require logical gates that avoid exposing encoded information to unprotected physical-qubit decay channels.
Method
The proposal encodes qubits in Schrödinger cat states stabilized by two- or four-photon driven dissipative processes and manipulates them with engineered Hamiltonians.
Results
The paper provides universal gates on the encoded qubits, including single-qubit rotations and a two-qubit entangling gate, with superconducting-circuit schemes for implementation.
Takeaways & Limitations
Four-component cat qubits combined with continuous photon-number parity measurements can track and correct dominant single-photon-loss errors while suppressing photon-dephasing errors.
Takeaways & Limitations
Two-component cat qubits are not protected against dominant single-photon loss, which induces uncorrected logical bit flips.
Abstract
from arXiv · showhide
We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator. This proposal exploits multi-photon driven dissipative processes to encode quantum information in logical bases composed of Schrödinger cat states. More precisely, we consider two schemes. In a first scheme, a two-photon driven dissipative process is used to stabilize a logical qubit basis of two-component Schrödinger cat states. While such a scheme ensures a protection of the logical qubit against the photon dephasing errors, the prominent error channel of single-photon loss induces bit-flip type errors that cannot be corrected. Therefore, we consider a second scheme based on a four-photon driven dissipative process which leads to the choice of four-component Schrödinger cat states as the logical qubit. Such a logical qubit can be protected against single-photon loss by continuous photon number parity measurements. Next, applying some specific Hamiltonians, we provide a set of universal quantum gates on the encoded qubits of each of the two schemes. In particular, we illustrate how these operations can be rendered fault-tolerant with respect to various decoherence channels of participating quantum systems. Finally, we also propose experimental schemes based on quantum superconducting circuits and inspired by methods used in Josephson parametric amplification, which should allow to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.
1. Introduction
The paper develops hardware-efficient protected logical qubits by encoding information in a quantum harmonic oscillator and stabilizing Schrödinger cat states with driven dissipative processes. It extends this approach from protected memory to universal, potentially fault-tolerant computation and proposes superconducting-circuit implementations.
- Motivation: A quantum harmonic oscillator provides redundant encoding and protection using its infinite-dimensional Hilbert space without adding extra decay channels.Photon loss is identified as the dominant oscillator error, so photon-loss syndromes can be targeted directly.
- Encoding: The encoding uses multi-component coherent-state superpositions, which become quasi-orthogonal for sufficiently large amplitude and protect information against photon loss events.For α = 2, the overlap between |α⟩ and |iα⟩ is below 10^-3 in the paper’s simulations.
- Protection: Photon-number parity measurements indicate photon-loss events, while deterministic amplitude decay requires energy repumping before coherent states substantially overlap.The parity expectation alternates as ⟨ψ(n)|Π|ψ(n)⟩ = (−1)^n.
- Dissipative protection: Two-photon dissipation protects two-component cat-state qubits against photon dephasing, but single-photon loss remains an uncorrected dominant error.The paper therefore extends the scheme to four-photon dissipation and four-component cat states, with continuous parity measurements tracking single-photon loss.
- Universal gates: Specific Hamiltonians enable arbitrary X-axis rotations and two-qubit entangling gates, while Kerr interactions supply π/2 rotations around the Y or Z axis for universality.The strong dissipative process projects evolution into the logical qubit’s degenerate, decoherence-free subspace through quantum Zeno dynamics.
- Implementation: The proposal includes circuit-QED schemes inspired by Josephson parametric amplification to engineer the required dissipative processes and Hamiltonians efficiently.The fixed two-cavity setup is designed to provide flexibility over the needed Hamiltonians and damping operator.
2. Driven dissipative multi-photon processes and protected logical qubits
Two- and four-photon driven dissipative processes stabilize cat-state manifolds with distinct protection properties. The four-photon scheme combines suppression of dephasing-induced phase flips with continuous parity-based tracking of single-photon loss.
- Two-photon driven dissipative process: Two-photon driving preserves photon-number parity and stabilizes even or odd Schrödinger cat states from corresponding parity initial states.Vacuum evolves to a pure even cat, while an odd-parity Fock state evolves to a pure odd cat.
- Two-photon driven dissipative process: The two-photon scheme encodes a logical qubit in even and odd cat states and robustly preserves its Bloch-sphere X components against dephasing.The logical phase-flip rate induced by dephasing is exponentially suppressed as the cat size increases.
- Two-photon driven dissipative process: Single-photon loss remains unprotected in the two-photon encoding, producing logical bit flips at rate |α|2κ1ph.The annihilation operator maps between the even and odd cat states, and the two-photon process does not suppress these jumps.
- Four-photon driven dissipative process: A four-photon process stabilizes a four-dimensional manifold spanned by coherent states |±α⟩ and |±iα⟩, while conserving photon number modulo 4.Continuous parity monitoring restricts the dynamics to even states, yielding a logical basis of four-component cat states.
3. Universal gates and fault-tolerance
The paper develops universal gates on dynamically protected cat-qubit encodings using quantum Zeno dynamics, including arbitrary X rotations, entangling gates, and a Kerr-based Z rotation. Simulations show high gate fidelities and preservation against photon loss and dephasing under the four-photon scheme.
- The gate set comprises arbitrary single-qubit X rotations, a fixed π/2 rotation around Z, and a two-qubit entangling gate.Together, these operations provide universal quantum computation on the encoded qubits.
- 3.1. Quantum Zeno dynamics for arbitrary rotations of a single qubit: Continuous multi-photon dissipation acts as a quantum-Zeno projection while weak driving fields implement arbitrary logical X rotations.For the two-photon scheme, displacement and re-projection produce the logical rotation; the same strategy extends to the four-photon encoding.
- 3.1. Quantum Zeno dynamics for arbitrary rotations of a single qubit: The simulated two-photon X rotations show only slight decay, while a time-scale ratio of 1/20 yields gate fidelities above 99.5%.The decay originates from higher-order terms and can be reduced by using a smaller drive-to-dissipation ratio, at the cost of longer gates.
- 3.2. Quantum Zeno dynamics for a two-qubit entangling gate: A projected beam-splitter interaction generates two-mode entanglement reaching Bell states, with fidelity above 99% at a moderate time-scale ratio of 1/20.Higher-order terms in the projected interaction account for the observed fidelity decay.
- 3.3. Kerr effect for π/2-rotation around Z-axis: A Kerr Hamiltonian supplies the fixed single-qubit rotation around Z needed to complete the universal gate set.For the four-component encoding, the operation realizes a −π/2 rotation around the logical Z axis.
- 3.4. Fault-tolerance: Under the four-photon process, continuous parity measurements preserve protection against single-photon loss during X rotations and entangling operations.The photon-loss error rate does not increase during these operations, while the four-photon process also protects against photon dephasing.
- 3.4. Fault-tolerance: With κ1ph = κ4ph/200, the probability of more than one photon-loss jump during an operation remains within 1%.The simulations indicate that a parity measurement after the operation may suffice for a significant coherence-time improvement under these parameters.
4. Towards an experimental realization within a circuit QED framework
The proposed circuit-QED architectures engineer two-photon dissipative stabilization and the Hamiltonians required for universal logical operations using coupled cavities, Josephson junctions, drives, and controlled dissipation. Numerical comparisons support the reduced model, while four-photon implementation remains subject to ongoing work.
- Two-photon driven dissipative process: A Josephson-junction architecture combines coupled cavities, coherent drives, and single-photon dissipation to realize the two-photon driven dissipative dynamics.The design selects nonlinear interaction terms through pump frequencies and uses lossy mode b to generate the effective dynamics of storage mode a.
- Two-photon driven dissipative process: Setting ωp = 2˜ωa − ˜ωb selects an interaction a^2b† + c.c., while driving and damping mode b produce the desired dynamics in mode a.Mode a stores and protects the quantum information.
- Model reduction: Under g2ph, ϵb ≪ κb, adiabatic elimination of mode b yields the reduced model of Eq. (1), whose fidelity dynamics converge comparably to the complete model.The remaining discrepancy is attributed to finite time-scale separation and nonzero Kerr and cross-Kerr terms.
- Universal logical gates: Resonant driving of mode a implements arbitrary logical X rotations, while a pumped Josephson-junction coupling provides a two-qubit entangling interaction.The entangling architecture couples two high-Q storage cavities through a central junction and uses a pump at ωZZ = (˜ωa1 − ˜ωa2)/2.
- Universal logical gates: A self-Kerr interaction implements a π/2 logical Z rotation by turning off pumping drives and waiting for π/χaa.The proposed circuit elements also support engineering the Hamiltonians needed for the logical gates.
- Four-photon extension: The four-photon process can be reduced by adiabatically eliminating mode b when g4ph, ϵb ≪ κb, but an architecture for realizing it is still ongoing work.The same cavity architecture with ωp = 4˜ωa − ˜ωb is proposed to induce the required four-photon coupling.
Appendix A.1. Asymptotic state for arbitrary initial state of the two-photon process
For arbitrary initial states, the two-photon driven dissipative process converges to an asymptotic density matrix supported on the two-component cat-state manifold. Conserved quantities determine the resulting logical-state populations and coherence.
- Asymptotic manifold: Every initial state converges exponentially to an asymptotic density matrix supported on the span of the two-component Schrödinger cat states.The asymptotic state may be mixed and is defined within the logical cat-state manifold.
- Conserved quantities: The population and coherence of the asymptotic logical state are determined by conserved quantities J++ and J+− evaluated on the initial state.These quantities remain sufficient to calculate the relevant logical degrees of freedom.
- Conserved quantities: J++ is conserved because the two-photon dynamics preserve photon-number parity.J++ acts as the positive-parity projector.
- Conserved quantities: The off-diagonal conserved quantity J+− extends the corresponding conserved quantity of the undriven dissipative two-photon process.The appendix establishes its conservation using the equations of motion and associated identities.
Appendix A.2. Asymptotic state for an initial coherent state of the two-photon process
For an initial coherent state, the appendix derives expressions for the asymptotic cat-state population and coherence using convergent Bessel-function sums and integral identities. The large-amplitude limits match the behavior shown numerically.
- Coherent-state initialization: For an initial coherent state |β⟩⟨β|, the asymptotic population c++ and coherence are obtained from the conserved quantities.The resulting expressions depend on the coherent-state amplitude and phase.
- Analytical derivation: The derivation converts a convergent Bessel-function sum into an integral by applying an addition theorem and interchanging the sum and integral.The remaining sum is evaluated as a Fourier series before a change of variables yields the final expression.
- Limiting behavior: The α = 0 limit reproduces the corresponding undriven result, while large-|β|^2 limits analytically corroborate Figure 2.The appendix identifies the two-photon system with a classical double-well system in the combined large-α, β regime.
Appendix A.3. Influence of dephasing on the two-photon process
Photon dephasing preserves the logical populations while causing only phase-flip errors, whose rate is exponentially suppressed as the cat-state photon number increases. The appendix’s analytical estimate agrees with numerical simulations in the small-dephasing regime.
- Logical effect of dephasing: Photon dephasing preserves the positive-parity population and leaves the logical |±Z⟩ populations unchanged.The logical coherence decays exponentially at first order, so dephasing induces phase-flip rather than bit-flip errors.
- Logical error rate: The logical phase-flip rate is exponentially suppressed with increasing cat-state photon number |α|^2.The rate is obtained from the first-order perturbative correction to the asymptotic manifold.
- Numerical validation: For small κφ/κ2ph, the numerical phase-flip eigenvalue approaches the analytical estimate.Figure A1 shows exponential convergence of the eigenvalue toward zero as |α| increases.
- Logical-state deformation: Dephasing slightly smears the phase-space peaks of the cat states but does not affect the encoded information represented by J++ and J+−.The basis elements acquire a small random phase under ensemble averaging.
Appendix A.4. Asymptotic behavior of the four-photon process
The four-photon process has an asymptotic manifold defined on a four-dimensional cat-state Hilbert space, with parity tracking restricting the dynamics. Numerical analysis finds photon dephasing suppresses the induced logical phase-flip rate nearly as in the two-photon process, apart from a slight delay.
- The four-photon process asymptotically occupies density matrices defined on a four-dimensional Hilbert space spanned by four-component cat states.
- Tracking parity restricts the dynamics to a subspace spanned by selected cat states.
- One conserved population quantity matches the non-driven case and is expressed as J00 = P∞_{n=0}|4n⟩⟨4n|.
- An analytical expression for the other conserved quantity, J02, remains unavailable, so photon-dephasing effects are analyzed numerically.
- The induced logical phase-flip rate shows nearly the same exponential suppression as in the two-photon process, with a slight delay.