Quantum Computing with Rotation-Symmetric Bosonic Codes

Abstract

Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code-agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on concatenation of number-phase codes and Bacon-Shor subsystem codes.I.

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

“…In some cases, the redundancy of the full infinite-dimensional Hilbert space can even be leveraged to detect and correct CV errors -random Gaussian displacements, rotations, and photon loss, for a few -without destroying the encoded information. Examples of bosonic codes include GKP [20], dual-rail [3,54], cat [55,56], hypercat [57,58], binomial [59], and general rotation-symmetric codes [60].…”

Photon-Number-Dependent Hamiltonian Engineering for Cavities

“…Continuous rotation of the encoded logical qubit around the Z axis can generate the whole family of phase shift gates R θ , including π/8 gate and Z gate, which are common elements of single-qubit gates for universal quantum computing [30]. For quantum information encoded in rotational-symmetric bosonic code that can correct up to (d n − 1)-photon loss errors [9], the logical states are…”

“…Tremendous experimental progress has been made in building highcoherence microwave photon cavities in circuit quantum electrodynamics (cQED) platforms [1][2][3][4]. The infinitedimensional Hilbert space of a single resonator enables flexible and hardware-efficient design of quantum error correction codes [5][6][7][8][9][10] and has led to the success in extending the logical qubit lifetime [11]. Controllable cavity systems can also be used to emulate the dynamics of the classically intractable many-body quantum systems due to their rapidly growing Hilbert space [12,13].…”

Encoding a qubit in a molecule

“…The analogous two-rotor "conditional-phase" operator, cphs ϕ = e −iϕL⊗L {cf. [131], Eq. (23)}, commutes withX φ ⊗ 1 and 1 ⊗X φ , but mapŝ…”

“…Now, the corrupted position label corresponding to codeword r can stray into the the Voronoi cell of some other element r = r. The above recovery will snap such error words to the wrong codewords, leading to logical errors. In the case of nonabelian codes, there may be errors due to the effect of the compensating element on µν (131).…”