Session B30: Quantum Error Correction and Quantum Control

Fault-tolerant quantum computation with asymmetric Bacon-Shor codes

Bacon-Shor codes are quantum subsystem codes which are constructed by combining together two quantum repetition codes, one protecting against $Z$ (phase) errors and the other protecting against $X$ (bit flip) errors. In many situations, for example flux qubits, the noise is biased such that faults that produce $Z$ errors are much more common than faults that produce $X$ errors; in these cases it is natural to consider an asymmetric Bacon-Shor code where the code protecting against $Z$ errors is longer than the code protecting against $X$ errors. This work describes fault-tolerant constructions for gadgets that achieve universal fault-tolerant quantum computation using asymmetric Bacon-Shor codes. Gadgets take advantage of the Bacon-Shor structure by breaking up into parallel smaller gadgets that act on a single row or column, with majority voting of the separate results. For a bias of $ \epsilon/\epsilon' = 10^{4}$, we prove a threshold around $2.5 \times 10^{-3}$. The effective error strength is shown to decrease rapidly (faster than polynomial) with decreasing $\epsilon$. Therefore it may be practical to use Bacon-Shor codes directly with no additional concatenation. This could greatly reduce the resource overhead required for fault-tolerant computation with biased noise.   

Quantum Error Correction: Optimal, Robust, or Adaptive? Or, Where is The Quantum Flyball Governor?

In \emph{The Human Use of Human Beings: Cybernetics and Society} (1950), Norbert Wiener introduces feedback control in this way: \begin{quote} \begin{small} ``This control of a machine on the basis of its actual performance rather than its expected performance is known as \emph{feedback} ... It is the function of control ... to produce a temporary and local reversal of the normal direction of entropy.'' \end{small} \end{quote} \emph{The} classic classroom example of feedback control is the all-mechanical flyball governor used by James Watt in the 18th century to regulate the speed of rotating steam engines. What is it that is so compelling about this apparatus? First, it is easy to understand how it regulates the speed of a rotating steam engine. Secondly, and perhaps more importantly, \emph{it is a part of the device itself}. A naive observer would not distinguish this mechanical piece from all the rest. So it is natural to ask, where is the \emph{all-quantum} device which is self regulating, ie, the Quantum Flyball Governor? Is the goal of quantum error correction (QEC) to design such a device? Devloping the computational and mathematical tools to design this device is the topic of this talk.   

Unified approach to approximate quantum error correction via the transpose channel

Much of the existing work on error correction focuses on the standard paradigm of perfect quantum error correction(QEC), where the recovery operation perfectly reverses the effects of a noise channel. Recent studies on approximate QEC(AQEC) have demonstrated possible advantages that arise from relaxing the requirement for perfect correction. However, while the recovery operation for perfectly correctable codes is well-known, finding the recovery for approximately correctable codes often requires difficult numerical procedures. We demonstrate an analytical, universal and near-optimal recovery map-the transpose channel- for AQEC codes, with optimality defined in terms of the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the QEC conditions and generalize them to a set of conditions for AQEC codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. Our results can also be extended to the general case of approximate operator quantum error correction, thus bringing us closer to a unified, analytical framework for AQEC.(Ref:PRA,81,062342(2010))   

Design of additive quantum codes via the codeword-stabilized framework

Codeword stabilized (CWS) construction defines a quantum code by combining a classical binary code with some underlying graph state. In general, CWS codes are non-additive but become additive stabilizer codes if derived from a linear binary code. Generic CWS codes typically require complex error correction; however, we show that the CWS framework is an efficient tool for constructing good stabilizer codes with simple decoding. We start by proving the lower Gilbert-Varshamov bound on the parameters of an additive CWS code which can be obtained from a given graph. We also show that cyclic additive CWS codes belong to a previously overlooked family of single-generator cyclic stabilizer codes; these codes are derived from a circulant graph and a cyclic binary code. Finally, we present several families of simple stabilizer codes with relatively good parameters, including a family of the smallest toric-like cyclic CWS codes which have length, dimension, and distance as follows: $[[t^2+(t+1)^2,1,2t+1]]$, $t=1,2, \ldots$ \\[4pt] [1] A. A. Kovalev, I. Dumer, and L. P. Pryadko, preprint arXiv:1108.5490 (2011).   

Error-threshold for topological subsystem quantum error-correcting codes

In general, stability against noise in a quantum computer can be achieved by storing quantum information redundantly. For instance, topological quantum error correction averts decoherence effects by encoding qubits in non-local degrees of freedom, while actively correcting for local errors. The key merit of these topological stabilizer codes lies in the intrinsic locality of the operations for syndrome measurement and error correction. Topological subsystem codes further facilitate practical applications by requiring only measurements of adjacent qubit pairs. We numerically determine the error threshold of topological subsystem codes for the depolarizing channel by mapping the problem onto a classical statistical spin model with bond disorder, which is analyzed via large-scale Monte Carlo simulations. In this picture, faulty qubits correspond to antiferromagnetic interactions between classical spins and the point in the disorder--temperature phase diagram where ferromagnetic order is lost corresponds to the error threshold of the underlying quantum bit system.   

Correlated Errors in the Surface Code

A milestone step into the development of quantum information technology would be the ability to design and operate a reliable quantum memory. The greatest obstacle to create such a device has been decoherence due to the unavoidable interaction between the quantum system and its environment. Quantum Error Correction is therefore an essential ingredient to any quantum computing information device. A great deal of attention has been given to surface codes, since it has very good scaling properties. In this seminar, we discuss the time evolution of a qubit encoded in the logical basis of a surface code. The system is interacting with a bosonic environment at zero temperature. Our results show how much spatial and time correlations can be detrimental to the efficiency of the code.   

Quantum Circuits for Measuring Levin-Wen Operators

We give explicit quantum circuits (expressed in terms of Toffoli gates, CNOTs and single qubit rotations) which can be used to perform quantum non-demolition measurements of the commuting set of vertex and plaquette operators that appear in the Levin-Wen model [1] for the case of doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error correcting code defined by the ground states of the Levin-Wen model --- a scenario envisioned in [2]. A key component in our construction is a quantum circuit ${\cal F}$ that acts on 5 qubits at a time and carries out a so-called $F$-move, a unitary operation whose form is essentially fixed by a self-consistency condition known as the pentagon equation. In addition to our measurement circuits we also give an explicit 7 qubit circuit which can be used to verify that ${\cal F}$ satisfies the full pentagon equation as well as a simpler 2 qubit circuit which verifies the essential nontrivial content of this equation. \newline [1] M.A. Levin and X.-G. Wen, Phys. Rev. B {\bf 71} 045110 (2005). \newline [2] R. Koenig, G. Kuperberg, and B.W. Reichardt, Ann. Phys {\bf 325}, 2707 (2010).   

Exchange-Only Dynamical Decoupling in the 3-Qubit Decoherence Free Subsystem

The Uhrig dynamical decoupling sequence achieves high-order decoupling of a single system qubit from its dephasing bath through the use of bang-bang Pauli pulses at appropriately timed intervals [1]. This high-order decoupling property of the Uhrig sequence has been extended to decouple general noise from single [2] and multiple [3] qubit systems, using single-qubit Pauli pulses. For the 3-qubit decoherence free subsystem (DFS) and related subsystem encodings, Pauli pulses are not naturally available operations; instead, exchange interactions provide all required encoded operations [4]. Here we demonstrate that exchange interactions alone can achieve high-order decoupling against general noise in the 3-qubit DFS. We present decoupling sequences for a 3-qubit DFS coupled to classical and quantum baths and evaluate the performance of the sequences through numerical simulations. References: [1] G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007). [2] J. R. West, B. H. Fong, and D. A. Lidar, Phys. Rev. Lett. 104, 130501 (2010). [3] Z.-Y. Wang and R.-B. Liu, Phys. Rev. A 83, 022306 (2011). [4] J. Kempe et al., Phys. Rev. A 63, 042307 (2001).   

Upper bounds on coherence preservation in dynamical decoupling

We explore the fundamental limits on coherence preservation by dynamical decoupling in terms of control time scales and the spectral bandwidth of the environment. Our main focus is a decohering qubit controlled by arbitrary sequences of $\pi$ pulses. Using methods from mathematical analysis, we establish a non-perturbative lower bound for the coherence loss in terms of the minimum pulse separation and the cutoff frequency of the environment. We argue that similar bounds are applicable to a variety of open-loop unitary control methods while we find no explicit dependence of such lower bounds on the total control time. We use these findings to generate bandwidth adapted dynamical decoupling sequences that can preserve a qubit up to arbitrary long times within the best fidelities theoretically possible given the available control resources. Our analysis reinforces the impossibility of fault-tolerance accuracy thresholds under purely reversible error control.   

Generalized Uhrig Dynamical Decoupling for Multi-Level Quantum Systems

Dynamical decoupling can efficiently suppress decoherence induced by the system-environment interaction. Recently, Uhrig proposed an efficient dynamical decoupling scheme, which uses only $N$ pulses to suppress dephasing noise to $O(T^{N+1})$ for a qubit system with total time evolution $T$. We generalize Uhrig's dynamical decoupling scheme from 2-level to $L$-level quantum systems. We find that $M=(L-1)N$ pulses are sufficient to suppress dephasing noise to $O(T^{N+1})$. We observe interesting patterns in the timing of these pulses, which depend on both $L$ and $N$ with various asymptotic forms for large $L$ or large $N$.   

Universal Dynamical Decoupling and Quantum Walks in Functional Spaces

We investigate the universal dynamical decoupling (DD) schemes, which can restore the coherence of quantum system independent of the details of system-environment interaction. We introduce a general mapping between DD sequences and quantum walks in functional spaces, and use it to prove the universality of various DD schemes such as quadratic DD, nested Uhrig DD, and Uhrig concatenated DD, as well as previously known universal schemes of concatenated DD, Uhrig DD and concatenated Uhrig DD.   

Protecting adiabatic quantum computation by dynamical decoupling

Adiabatic quantum computation (AQC) relies heavily on a systems ability to remain in its ground state with high probability throughout the entirety of the adiabatic evolution. System-environment interactions present during the evolution manifest decoherence, thereby increases the probability of excitation. In this work, it is shown that the existence of such noise-producing terms can be dramatically reduced by Dynamical Decoupling (DD). In particular, we consider a multi-qubit system subjected to a classical bath modeled by random Gaussian-correlated noise. The performance of deterministic schemes such as Concatenated Dynamical Decoupling (CDD) and Nested Uhrig Dynamical Decoupling (NUDD) are analyzed for Grover's search algorithm and the two-qubit Satisfiability (2-SAT) problem. The CDD evolution substantially increases noise suppression with increasing concatenation level. In contrast, improvements in performance are only observed for specific sequence orders in the NUDD scheme. These results are verified for both adiabatic evolutions in terms of the total adiabatic run time and minimum pulse interval.   

Polar codes for achieving the classical capacity of a quantum channel

We construct the first near-explicit, linear, polar codes that achieve the capacity for classical communication over quantum channels. The codes exploit the channel polarization phenomenon observed by Arikan for classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by ``channel combining'' and ``channel splitting,'' in which a fraction of the synthesized channels is perfect for data transmission while the other fraction is completely useless for data transmission, with the good fraction equal to the capacity of the channel. Our main technical contributions are threefold. First, we demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity is O(N log N), where N is the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, i.e., the word error rate decays exponentially with the blocklength of the code. For a quantum channel with binary pure-state outputs, such as a binary-phase-shift-keyed coherent-state optical communication alphabet, the symmetric Holevo information rate is in fact the ultimate channel capacity, which is achieved by our polar code.   

Improved coded optical communication error rates using joint detection receivers

It is now known that coherent state (laser light) modulation is sufficient to reach the ultimate quantum limit (the Holevo bound) for classical communication capacity. However, all current optical communication systems are fundamentally limited in capacity because they perform measurements on single symbols at a time. To reach the Holevo bound, joint quantum measurements over long symbol blocks will be required. We recently proposed and demonstrated the ``conditional pulse nulling'' (CPN) receiver -- which acts jointly on the time slots of a pulse-position-modulation (PPM) codeword by employing pulse nulling and quantum feedforward -- and demonstrated a 2.3 dB improvement in error rate over direct detection (DD). In a communication system coded error rates are made arbitrary small by employing an outer code (such as Reed-Solomon (RS)). Here we analyze RS coding of PPM errors with both DD and CPN receivers and calculate the outer code length requirements. We find the improved PPM error rates with the CPN translates into $>$10 times improvement in the required outer code length at high rates. This advantage also translates increase the range for a given coding complexity. In addition, we present results for outer coded error rates of our recently proposed ``Green Machine'' which realizes a joint detection advantage for binary phase shift keyed (BPSK) modulation.