Session M27: Focus Session: Quantum Error Correction and Decoherence Control I

Dynamical quantum error correction: recent achievements and prospects

Precisely controlling the dynamics of real-world open quantum systems is a central challenge across quantum science and technology, with implications ranging from quantum physics and chemistry to fault-tolerant quantum computation. Dynamical quantum error correction strategies based on open-loop time-dependent modulation of the system dynamics provide a general perturbative framework for boosting physical-layer fidelities in the non-Markovian regime. I will describe recent progress in designing dynamical error correction schemes able to incorporate various system and control constraints encountered in realistic scenarios. In particular, I will show how to employ dynamical decoupling methods to achieve high-fidelity quantum storage for long times, while minimizing access latency and sequencing complexity, and how to synthesize dynamically corrected quantum gates for simultaneously canceling non-Markovian decoherence and control errors, while accommodating internal always-on dynamics and limited control. In the process, I will make contact with current qubit devices to the extent possible and point to remaining challenges and directions for further explorations.   

Quantum Control and Fault-tolerance

Quantum control (QC) and the methods of fault-tolerant quantum computing (FTQC) are two of the cornerstones on which the hope for a quantum computer rests. However QC methods do not generally scale well with the size of the system, and it is not known how their performance is hindered when integration with FTQC methods, especially considering these demand a large system size overhead, is attempted under realistic noise models. Here we study this problem using dynamical decoupling in the bang-bang limit as a toy model, with a non-Markovian noise where interactions decay with distance, and show that there exists a regime of the norms of the relevant Hamiltonians, in which dynamical decoupling protected gates provide an advantage over the bare gate implementation. This is a first step towards showing that QC protocols designed for a small set of qubits can be extended to larger sets without a significant loss of performance, as long as the noise model behaves reasonably well.   

Preserving electron spin coherence by dynamical decoupling based on Nitrogen-Vacancy center in diamond

To exploit the quantum coherence of electron spins in solids in future technologies such as quantum manipulating, it's first vital to overcome the problem of spin decoherence due to their coupling the noisy environment. Dynamical decoupling is a particularly promising strategy for combating decoherence. I will briefly introduce the roadmap for dynamical decoupling and show our experimental research on the field in detail. We first applied the optimal dynamical decoupling scheme [1] on electron spins of ensemble sample [2]. Based on the technology, the dynamical decoupling sequence was used to observe the anomalous coherence effect and of single electron spin based on nitrogen-vacancy defect center in diamond [3]. For application, combined the dynamical decoupling together with quantum metrology protocol, the phase estimation was enhanced [4]. Instead of pulsed model, continuous dynamical decoupling was realized in our experiment and applied to protect quantum gate [5]. The next step, we will apply multi flip pulses to enhance the magnetic field sensitivity of NV center towards to the micro-scale magnetic resonance and single molecular imaging. [1] G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007) [2] J. Du, et al., Nature 461, 1265 (2009) [3] P. Huang, et al., Nature Communications, 2, 570 (2011) [4] X. Rong, et al., Europhys. Lett. 95, 60005 (2011) [5] X. Xu, et al., Phys. Rev. Lett. 109, 070502 (2012)   

Improving quantum gate fidelities by using a qubit to measure microwave pulse distortions

We present a new method for determining pulse imperfections and improving the single-gate fidelity in a superconducting qubit. By applying consecutive positive and negative $\pi$ pulses, we amplify the qubit evolution due to microwave pulse distortions, which causes the qubit state to rotate around an axis perpendicular to the intended rotation axis. Measuring these rotations as a function of pulse period allows us to reconstruct the shape of the microwave pulse arriving at the sample. Using the extracted response to predistort the input signal, we are able to improve the pulse shapes and to reach an average single-qubit gate fidelity higher than $99.8\%$.   

Accurate quantum $Z$ rotations with less magic

We present quantum protocols for executing arbitrarily accurate $\pi/2^k$ rotations of a qubit about its $Z$ axis. Unlike reduced instruction set computing (RISC) protocols which use a two-step process of synthesizing high-fidelity ``magic'' states from which $T = Z(\pi/4)$ gates can be teleported and then compiling a sequence of adaptive stabilizer operations and $T$ gates to approximate $Z(\pi/2^k)$, our complex instruction set computing (CISC) protocol distills magic states for the $Z(\pi/2^k)$ gates directly. Replacing this two-step process with a single step results in substantial reductions in the number of gates needed. The key to our construction is a family of shortened quantum Reed-Muller codes of length $2^{k+2}-1$, whose distillation threshold shrinks with $k$ but is greater than 0.85\% for $k \leq 6$. AJL and CC were supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Fault-tolerant, nondestructive measurement of logical operators and quantum teleportation in large stabilizer codes

Fault-tolerant quantum computation seeks to perform large calculations by protecting quantum information against decoherence using quantum error-correcting codes. Such schemes have been widely studied, but the resources needed to actually perform a fault-tolerant computation are daunting. In principle, it may be possible to reduce this overhead by using large block codes with significantly higher rates. Logical gates can be done in such a scheme by teleporting the logical qubits between code blocks. Logical teleportation can be done fault-tolerantly by measuring a particular set of logical operators. This measurement involves preparing an entangled ancillary state and doing a transversal circuit between the codeword and the ancilla. We study this procedure, and show that a wide range of such measurement protocols exist. There is a trade-off between the size of of the ancilla and the robustness against errors; for a large codeword, it may be fruitful to use a larger ancilla that has greater error-correcting power.   

Quantum ``hyperbicycle'' low-stabilizer-weight finite-rate error correction codes

We construct a large family of finite-rate quantum error correcting codes (QECCs) which interpolate between the hypergraph-product [1] and generalized bicycle codes [2]. The construction allows for the lower and upper bounds on the distance which generally scale as a square root of the block size; in several important cases the two bounds coincide. The constructed QECCs include several new classes of codes with low stabilizer weights; they can offer an advantage compared to the toric codes. \\[0.2em] [1] J.-P. Tillich and G. Z\'emor, in Proc. IEEE Int. Symp. Inf. Th., 2009 (ISIT 2009), pp. 799-803.\\[0em] [2] D. MacKay, G. Mitchison, and P. McFadden, IEEE Trans. on Inf. Th., {\bf 50}, 2315 (2004).   

Error correction with quantum low-density parity check codes

We study quantum low-density parity check (LDPC) codes and their fault tolerance. We show that any family of quantum LDPC codes where each syndrome measurement involves a limited number of qubits, and each qubit is involved in a limited number of measurements (as well as any similarly-limited family of classical LDPC codes), where distance scales as a positive power of the number of physical qubits, has a finite error probability threshold. We conclude that for sufficiently large quantum computers, finite-rate quantum LDPC codes can offer an advantage over the toric codes. Error correction in the presence of errors in syndrome measurements is also addressed. We discuss possible realizations of decoders and their error thresholds, e.g. in relation to LDPC versions of the quantum hypergraph-product codes [1] and their generalizations [2].\\[4pt] [1] J.P. Tillich, G. Zemor, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 799 (2009). \newline [2] A. A. Kovalev and L. P. Pryadko, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 348 (2012).   

Relative performance of ancilla verification and decoding in the [[7,1,3]] Steane code

We present numerical simulation results comparing the logical error rates for the fault-tolerant [[7,1,3]] Steane code using standard ancilla verifications techniques vs. the newer method of ancilla decoding, as described in [1]. We simulate a realistic QEC procedure in which failed ancilla creation requires storing the data until a new ancilla can be created; we expect the decoding method, which avoids the need for such storage, to be advantageous when the failure probability is sufficiently high. For the [[7,1,3]] code, we analyze the effect of both different syndrome extraction techniques and of different classes of physical error (initialization, measurement, hold etc.) on the relative performance of these two methods.\\ $\bf 1$. David P. DiVincenzo and Panos Aliferis, Phy. Rev. Lett. $\bf 98$ 020501(2007).   

Quantum Error Correction with Mixed State Ancilla Qubits

It is commonly assumed that ancilla qubits must be in a pure state for successful quantum error correction. We show in our talk that they can initially be in any mixed state if the error operator acts simultaneously on all the physical qubits (fully correlated noise). In particular, they can be in the uniformly mixed state, which makes implementation of our scheme extremely cheap. We also note that 1-qubit gate operations can be implemented easily within the codeword. We experimentally demonstrated our scheme by using a liquid state NMR quantum computer. The encoded state has an interesting nature in terms of quantum discord, which is purely quantum correlations between the data qubit and the ancilla qubits.