Design of nanophotonic circuits for autonomous subsystem quantum error correction

We reapply our approach to designing nanophotonic quantum memories to formulate an optical network that autonomously protects a single logical qubit against arbitrary single-qubit errors. Emulating the 9 qubit Bacon-Shor subsystem code, the network replaces the traditionally discrete syndrome measurement and correction steps by continuous, time-independent optical interactions and coherent feedback of unitarily processed optical fields. PACS numbers: 03.67.Pp, 42.50.Ex, 42.50.Pq, 02.30.Yy Submitted to: New J. Phys. Design of nanophotonic circuits for autonomous subsystem quantum error correction 2 Traditional approaches to designing quantum memories presume a complex classical apparatus operating in parallel with the memory to detect and correct errors as they arise [1, 2]. Typically, the mechanics of this essential half of the memory system are hardly considered at all, taking a back seat in the analysis to the qubits in which the information is stored. This assumption of a perfect, classical overseer glosses over the inherent technological mismatch of typically fast, nanoscale and cold quantum systems with slow, mesoscopic, and/or hot classical ones. The extreme performance demands for quantum information processing should motivate solutions that weigh classical and quantum resources wholistically and utilize system models as close to physical mechanics as possible. In [3] we described an approach to designing protected quantum memory devices naturally suited to nanophotonic implementations that is technologically homogeneous and requires no “off-chip” oversight, simply “power” in the form of cw laser inputs. The particular design example in [3] emulates the bitor phase-flip quantum error correcting (QEC) code [1, 2] with a single logical qubit stored in the collective state of three multilevel “atoms,” each strongly coupled to three different optical resonators. Two more cavity quantum electrodynamical (cQED) systems serve as the binary “controllers” [4] for these qubits. These five cQED devices are connected by a network of singlemode waveguides and beamsplitters that when appropriately powered by cw laser inputs continually and simultaneously cause the internal states of the controllers to reflect the joint parities of the qubits and corrective feedback on the qubits to be implemented according to the controller states. Essentially a bit-/phase-flip QEC code, the design in [3] can only protect the logical qubit against either single-qubit bit-flip or single-qubit phase-flip errors, depending on the configuration. To emphasize how straightforwardly the approach scales to more powerful QEC codes, in this article we describe, model and simulate an autonomous nanophotonic network that emulates the 9 qubit BaconShor subsystem code [5, 6], the smallest of a class of naturally fault-tolerant QEC codes capable of protecting a single logical qubit from arbitrary single-qubit errors (see Appendix A for a description of this code). While these networks are conceived of intuitively, our analysis arises from a mathematical framework of open quantum optical systems that takes quantum field theory as its physical basis [7, 8] but most closely resembles a quantum generalization of electrical circuit theory [9, 10]. Although rigorous stochastic dynamical modeling is often unfamiliar to physicists, this formalism [11] is very physically intuitive once internalized and sufficiently flexible to describe the continuous-time dynamics of most systems foreseeable in quantum optical networks. In our case, cQED devices are modeled as distinct Hamiltonian systems that couple weakly to free, bosonic fields that scatter off of each device in series and in parallel along paths set by the single-mode waveguides linking the network devices [9]. As our network operates “autonomously,” without any user monitoring necessary, the equation of motion that describes the closed loop time-evolution of the devices (i.e. the dynamics after tracing over the field degrees of freedom) is a deterministic master equation [8, 3]. Moreover, as all the Hamiltonian Design of nanophotonic circuits for autonomous subsystem quantum error correction 3