Operator quantum error correction

This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques - i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method - as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of unitarily noiseless subsystems.

Stabilizer formalism for operator quantum error correction

Operator quantum error correction is a recently developed theory that provides a generalized and unified framework for active error correction and passive error avoiding schemes. In this Letter, we describe these codes using the stabilizer formalism. This is achieved by adding a gauge group to stabilizer codes that defines an equivalence class between encoded states. Gauge transformations leave the encoded information unchanged; their effect is absorbed by virtual gauge qubits that do not carry useful information. We illustrate the construction by identifying a gauge symmetry in Shor's 9-qubit code that allows us to remove 3 of its 8 stabilizer generators, leading to a simpler decoding procedure and a wider class of logical operations without affecting its essential properties. This opens the path to possible improvements of the error threshold of fault-tolerant quantum computing.

Continuous quantum error correction by cooling

We describe an implementation of quantum error correction that operates continuously in time and requires no active interventions such as measurements or gates. The mechanism for carrying away the entropy introduced by errors is a cooling procedure. We evaluate the effectiveness of the scheme by simulation, and remark on its connections to some recently proposed error prevention procedures.

Quantum Error Correction via Codes over GF(2)

It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 371 length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.

Quantum error correction with biased noise

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security.

At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level.

In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations.

In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction.

In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.

Bound on quantum computation time: Quantum error correction in a critical environment

We obtain an upper bound on the time available for quantum computation for a given quantum computer and decohering environment with quantum error correction implemented. First, we derive an explicit quantum evolution operator for the logical qubits and show that it has the same form as that for the physical qubits but with a reduced coupling strength to the environment. Using this evolution operator, we find the trace distance between the real and ideal states of the logical qubits in two cases. For a super-Ohmic bath, the trace distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time before the trace distance exceeds a value set by the user. © 2010 The American Physical Society.

Entanglement production by quantum error correction in the presence of correlated environment

We analyze the effect of a quantum error correcting code on the entanglement of encoded logical qubits in the presence of a dephasing interaction with a correlated environment. Such correlated reservoir introduces entanglement between physical qubits. We show that for short times the quantum error correction interprets such entanglement as errors and suppresses it. However, for longer time, although quantum error correction is no longer able to correct errors, it enhances the rate of entanglement production due to the interaction with the environment.

Quantum-error correction on linear-nearest-neighbor qubit arrays

A quantum circuit implementing 5-qubit quantum-error correction on a linear-nearest-neighbor architecture is described. The canonical decomposition is used to construct fast and simple gates that incorporate the necessary swap operations allowing the circuit to achieve the same depth as the current least depth circuit. Simulations of the circuit's performance when subjected to discrete and continuous errors are presented. The relationship between the error rate of a physical qubit and that of a logical qubit is investigated with emphasis on determining the concatenated error correction threshold.

InP-based active and passive components for communication systems at 2 μm

Progress on advanced active and passive photonic components that are required for high-speed optical communications over hollow-core photonic bandgap fiber at wavelengths around 2 μm is described in this paper. Single-frequency lasers capable of operating at 10 Gb/s and covering a wide spectral range are realized. A comparison is made between waveguide and surface normal photodiodes with the latter showing good sensitivity up to 15 Gb/s. Passive waveguides, 90° optical hybrids, and arrayed waveguide grating with 100-GHz channel spacing are demonstrated on a large spot-size waveguide platform. Finally, a strong electro-optic effect using the quantum confined Stark effect in strain-balanced multiple quantum wells is demonstrated and used in a Mach-Zehnder modulator capable of operating at 10 Gb/s.

Evaluation of an inactivated Ross River virus vaccine in active and passive mouse immunization models and establishment of a correlate of protection

Ross River Virus has caused reported outbreaks of epidemic polyarthritis, a chronic debilitating disease associated with significant long-term morbidity in Australia and the Pacific region since the 1920s. To address this public health concern, a formalin- and UV-inactivated whole virus vaccine grown in animal protein-free cell culture was developed and tested in preclinical studies to evaluate immunogenicity and efficacy in animal models. After active immunizations, the vaccine dose-dependently induced antibodies and protected adult mice from viremia and interferon α/β receptor knock-out (IFN-α/βR(-/-)) mice from death and disease. In passive transfer studies, administration of human vaccinee sera followed by RRV challenge protected adult mice from viremia and young mice from development of arthritic signs similar to human RRV-induced disease. Based on the good correlation between antibody titers in human sera and protection of animals, a correlate of protection was defined. This is of particular importance for the evaluation of the vaccine because of the comparatively low annual incidence of RRV disease, which renders a classical efficacy trial impractical. Antibody-dependent enhancement of infection, did not occur in mice even at low to undetectable concentrations of vaccine-induced antibodies. Also, RRV vaccine-induced antibodies were partially cross-protective against infection with a related alphavirus, Chikungunya virus, and did not enhance infection. Based on these findings, the inactivated RRV vaccine is expected to be efficacious and protect humans from RRV disease