Session Y37b: Noise, Dynamical Decoupling, and Quantum Error Correction

Implementing small quantum codes with superconducting qubits

The ability to detect and correct errors is essential to any error correction protocol in quantum systems due to the fragile nature of quantum information. Demonstrating the encoding and decoding of logical states has become an important experimental pursuit. In particular, doing so fault-tolerantly will be critical to test the viability of different quantum error correction protocols. The surface code is particularly amenable to the implementation of error correcting protocols due to its use of nearest-neighbor interactions and relatively high error threshold. As a first step towards larger fault-tolerant circuits, we can study small codes with an accessible number of qubits in the current state-of-the art technology. The smallest code that detects a general error requires four data qubits. With one additional qubit as a syndrome qubit, we can study the [[4,2,2]] code and prepare one of its logical qubits fault-tolerantly. The device consists of five superconducting qubits arranged in a sublattice of the surface code, where a central syndrome qubit is coupled to four data qubits via bus resonators. I will present how these logical states are prepared and compare their lifetimes with each data qubit.   

Evanescent-Wave Johnson Noise in Small devices

In many quantum computer architectures, the qubits are in close proximity to metallic device elements. The fluctuating currents in the metal give rise to noisy electromagnetic fields that leak out into the surrounding region. These fields are known as evanescent-wave Johnson noise. The noise can decohere the qubits. A novel mapping of the quantum-mechanical problem onto a problem in classical electrodynamics simplifies the calculations. We present the general theory of this effect for charge qubits subject to electric noise and for spin and magnetic qubits subject to magnetic noise. New results are presented for the local noise spectral density in the vicinity of cylindrical conductors such as small antennae, noise from objects that can be treated as dipoles, and noise correlation functions for several geometries. We summarize the current state of the comparison of theory with experimental results on decoherence times of qubits.   

Using quantum process tomography to characterize decoherence in an analog electronic device

The mathematical structure of a universal gate-based quantum computer can be emulated faithfully on a classical electronic device using analog signals to represent a multi-qubit state. We describe a prototype device capable of performing a programmable sequence of single-qubit and controlled two-qubit gate operations on a pair of voltage signals representing the real and imaginary parts of a two-qubit quantum state. Analog filters and true-RMS voltage measurements are used to perform unitary and measurement gate operations. We characterize the degradation of the represented quantum state with successive gate operations by formally performing quantum process tomography to estimate the equivalent decoherence channel. Experimental measurements indicate that the performance of the device may be accurately modeled as an equivalent quantum operation closely resembling a depolarizing channel with a fidelity of over 99{\%}.   

A geometrical approach to dynamical decoupling with smooth pulses

In order to perform high-fidelity quantum information processing, reducing the effects of noise is an essential task. It is well known that a system can be decoupled from noise dynamically by using carefully designed pulse sequences based on delta-function or square waveforms such as spin echo or CPMG. However, such ideal pulses are often challenging to implement experimentally with high fidelity. We present an analytical approach that enables one to generate an unlimited number of smooth, experimentally feasible pulses that perform dynamical decoupling or dynamically corrected gates. Our method is based on a simple geometric picture that facilitates the identification of driving fields that cancel errors in the single-qubit evolution operator to second order or beyond. We demonstrate that this scheme can significantly enhance the fidelity of single-qubit gates in the case of noise with a 1/f power spectrum.   

Experimental realization of robust dynamical decoupling with bounded controls in a solid-state spin system

Dynamical decoupling is a powerful method to combat decoherence of quantum systems caused by coupling to slow-varying environment. Here We experimentally demonstrate a robust dynamical decoupling protocol with bounded controls using long soft pulses, eliminating a challenging requirement of strong control pulses in conventional implementations. This protocol is accomplished by designing the decoupling propagators to go through a Eulerian cycle of the coupler group [Phys. Rev. Lett. 90, 037901(2003)]. We use the solid-state spin qubits carried by the Nitrogen-Vacancy (NV) centers in a diamond as a testbed and demonstrate that this Eulerian decoupling scheme increases the coherence time by two orders of magnitude in our experiment under either dephasing or a universal noise environment.   

On the Effect of Coherence of Noise in Quantum Error Correction

We evaluated how the coherence of noise affects the error threshold under a coherent noise model. As the simplest model, we choose the one-dimensional (1D) quantum repetition code with repetitive parity measurements. Quantum error correction (QEC) with the 1D repetition code is simple but still able to capture a necessary ingredient for fault-tolerant QEC, and hence was experimentally demonstrated as a building block for scalable fault-tolerant quantum computation. We construct an efficient classical algorithm to simulate the quantum circuits for the QEC process with the 1D repetition code under a coherent noise model. The key idea of our algorithm is mapping all noise process and parity measurements into non-unitary free-fermionic dynamics. By using this algorithm, we calculated the error threshold and found that the existence of coherence in the noises significantly decrease the error threshold.   

Hard decoding algorithm for optimizing thresholds under general Markovian noise

With the advent of small scale quantum devices, studying the performance of quantum error correcting code's for realistic noise models is becoming increasingly important. In this work, we present an efficient hard decoding algorithm for optimizing thresholds of an error correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. Using our hard-decoding algorithm, we compute threshold values and error rates for a variety of error correcting code's. We show that thresholds for coherent noise can be significantly improved by exploiting transversal non-Pauli gates. Furthermore, the application of our hard decoding algorithm to coherent errors leads to better thresholds than if applied to the channel's Pauli-twirled counterpart. Lastly, we show that our optimized decoder is robust to perturbations about a noise model. Consequently, our decoder leads to reduced error rates even when applied to imperfectly characterized experimental noise.   

Performance analysis of fault-tolerant quantum error correction against non-Clifford errors

As error rates of quantum gates implemented in recent experiments approach a fault-tolerant threshold of a 2D planer surface code against a depolarizing noise model, it becomes more important to investigate performance of quantum error correction codes against more general and realistic noise models. A brute-force simulation for the investigation on a classical computer requires an exponential amount of memory, and we need alternative methods for the purpose. The standard approach assuming depolarizing (or Clifford) error models, which is not realistic, can overestimate the performance, and it is not valid to apply the results to experiments. On the other hand, a rigorous approach with the diamond norm is applicable to realistic error models but greatly underestimates the performance and is not practical. Here we propose a new theoretical framework for evaluating performances of quantum error correction, which is practical and applicable to a wider class of error models. We apply the method to a quantum 1D repetition code, and numerically evaluate the performance.   

Quantum supremacy of fault-tolerant quantum computation in a pre-threshold region

Demonstrating quantum supremacy, a complexity-guaranteed quantum advantage against over the best classical algorithms by using less universal quantum devices, is an important near-term milestone for quantum information processing. Here we develop a threshold theorem for quantum supremacy with noisy quantum circuits in the pre-threshold region, where quantum error correction does not work directly. By using the postselection argument, we show that the output sampled from the noisy quantum circuits cannot be simulated efficiently by classical computers based on a stable complexity theoretical conjecture, i.e., non-collapse of the polynomial hierarchy. By applying this to fault-tolerant quantum computation with the surface codes, we obtain the threshold value $2.84\%$ for quantum supremacy, which is much higher than the standard threshold $0.75\%$ for universal fault-tolerant quantum computation with the same circuit-level noise model. Moreover, contrast to the standard noise threshold, the origin of quantum supremacy in noisy quantum circuits is quite clear; the threshold is determined purely by the threshold of magic state distillation, which is essential to gain a quantum advantage.   

Experimental implementation of the Bacon-Shor code with 10 entangled photons

The number of qubits that can be effectively controlled in quantum experiments is growing, reaching a regime where small quantum error-correcting codes can be tested. The Bacon-Shor code is a simple quantum code that protects against the effect of an arbitrary single-qubit error. In this work, we propose an experimental implementation of said code in a post-selected linear optical setup, similar to the recently reported 10-photon GHZ generation experiment. In the procedure we propose, an arbitrary state is encoded into the protected Shor code subspace, and after undergoing a controlled single-qubit error, is successfully decoded.   

Noise Reduction in EIT Quantum Memories based Cs Atoms

Electromagnetically induced transparency (EIT) is a widely used approach for quantum memories. In an EIT-based quantum memory, a strong residual control beam comes out together with a read-out signal at single-photon level. The strong residual control beam becomes a main noise source in the system. Noise reduction becomes critical for the quantum memory because noise reduces the quantum information fidelity. For an operational EIT quantum memory, the strong residual power of the control beam must be greatly reduced. In an EIT quantum memory based on warm atoms, the signal and control beams propagate in the same direction, and with very small frequency difference, so noise reduction becomes a very challenging issue. To solve this problem, three types of filtration including a polarization filter, an F-P etalon filter and an optically pumped absorption atomic filter have been developed in our lab. The overall noise reduction reaches 125 dB, which satisfies the requirement of quantum memory applications. By using the developed filtration elements, our quantum memory successfully demonstrated storage and retrieval of quantum signals at a single photon level with high fidelity.   

Schemes of detecting nuclear spin correlations by dynamical decoupling based quantum sensing

Single-molecule sensitivity of nuclear magnetic resonance (NMR) and angstrom resolution of magnetic resonance imaging (MRI) are the highest challenges in magnetic microscopy. Recent development in dynamical decoupling (DD) enhanced diamond quantum sensing has enabled NMR of single nuclear spins and nanoscale NMR. Similar to conventional NMR and MRI, current DD-based quantum sensing utilizes the “frequency fingerprints” of target nuclear spins. Such schemes, however, cannot resolve different nuclear spins that have the same noise frequency or differentiate different types of correlations in nuclear spin clusters. Here we show that the first limitation can be overcome by using “wavefunction fingerprints” of target nuclear spins [1], which is much more sensitive than the "frequency fingerprints" to weak hyperfine interaction between the targets and a sensor, while the second one can be overcome by a new design of two-dimensional DD sequences composed of two sets of periodic DD sequences with different periods [2], which can be independently set to match two different transition frequencies. Our schemes not only offer an approach to breaking the resolution limit set by "frequency gradients" in conventional MRI, but also provide a standard approach to correlation spectroscopy for single-molecule NMR. References: [1] W. -L. Ma and R. -B. Liu, Phys. Rev. Appl. 6, 024019 (2016). [2] W. -L. Ma and R. -B. Liu, Phys. Rev. Appl. (in press), preprint arXiv:1512. 03548 (2015).   

Arbitrary n-Qubit State Transfer Using Coherent Control and Simplest Switchable Local Noise

We study the reachable sets of open n-qubit quantum systems, the coherent parts of which are under full unitary control, with time-modulable Markovian noise acting on a single qubit as an additional degree of incoherent control. In particular, adding bang-bang control of amplitude damping noise (non-unital) allows the dynamic system to act transitively on the entire set of density operators. This means one can transform any initial quantum state into any desired target state. Adding switchable bit-flip noise (unital), on the other hand, suffices to explore all states majorised by the initial state. We have extended our open-loop optimal control package DYNAMO to also handle incoherent control so that these unprecedented reachable sets can systematically be exploited in experiments. We propose implementation by a GMon, a superconducting device with fast tunable coupling to an open transmission line, and illustrate how open-loop control with noise switching can accomplish all state transfers without the need for measurement-based closed-loop feedback schemes with a resettable ancilla. Based on arXiv:1605.06473