Session F38: Focus Session: Quantum Error Correction II

Optimizing the frequency of post-selected quantum error correction~in the [[7,1,3]] Steane code

A common assumption in analyses of error thresholds and quantum computing in general is that one applies fault-tolerant quantum error correction (FTQEC) after every logical gate.~ This, however, is known not to always be optimal if the FTQEC procedure itself can introduce errors. We investigate the effect of varying the number of logical gates between FTQEC operations, and in particular the case where failure of a postselection condition in FTQEC may cause FTQEC to be skipped with high probability. By using a simplified model of errors induced in FTQEC, we derive an expression for the logical error rate as a function of error-correction frequency, and show that in this model the optimal frequency is insensitive to postselection failure probability for a large range of such probabilities. We compare the model to data derived from Monte Carlo simulation for the [[7,1,3]] Steane code.   

How Often Must We Apply Syndrome Measurements?

Quantum information is encoded into Quantum Error Correction codes to protect it from decoherence. The detection and correction of possible errors is done via syndrome measurements. Standard quantum fault tolerance approaches assume that syndrome measurements are applied after the implemenation of any gate. However, this is resource intensive utilizing much time and many qubits. In this talk we explore whether it is necessary to apply syndrome measurements so often. We examine different syndrome measurement techniques for the [[7,1,3]] code and compare the output state fidelity based on how often syndrome measurements are applied and the error environment. In this way we demonstrate the tradeoff between accuracy and resource consumption.   

Perfect Zeno effect through imperfect measurements at a finite frequency

The quantum Zeno effect (QZE) is usually thought to require infinitely frequent and perfect (i.e., projective) measurements to freeze the dynamics of quantum states. We show that perfect freezing of quantum states can also be achieved by more realistic non-projective measurements performed at a finite frequency. Furthermore, we show that, in the case of qubits, in contrast to the usual QZE, the state freezing via imperfect measurements can be adjusted to preserve arbitrary states in the Bloch sphere.   

Efficient estimation of quantum error correction thresholds in the presence of errors outside the Clifford group

Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code (QECC). It is common to model the noise as a depolarizing Pauli channel. However, it is not clear how sensitive a code's threshold is to the noise model, and whether or not a depolarizing channel is a good approximation for realistic errors. We have shown that, at the physical single-qubit level, efficient and more accurate approximations can be obtained\footnote{M. Gutierrez \textit{et al.}, Phys. Rev. A. \textbf{87}, 030302(R) (2013).}. We now examine the feasibility of employing these approximations to obtain better estimates of a QECC's threshold. We calculate the level-1 pseudo-threshold for the Steane [[7,1,3]] code for amplitude damping and dephasing along a non-Clifford axis. The expanded channels estimate the pseudo-threshold more accurately than the Pauli channels. However, at the logical level, the Pauli channels result in states that are closer to the states after the realistic errors, which is consistent with results by Geller and Zhou\footnote{M.R. Geller and Z. Zhou, Phys. Rev. A \textbf{88}, 012314 (2013).}. This suggests that employing Pauli channels can actually result in very accurate approximations of a QECC's performance.   

Logical Error Rate in the Pauli Twirling Approximation

Understanding how decoherence and intrinsic errors affect information processing is an important task in quantum computing. The Gottesman-Knill Theorem offers a class of error models that are efficiently simulable on a classical computer. The simplest of these error models, Pauli Twirling Approximation (PTA), which is got by twirling a completely positive channel over the Pauli basis is widely applied; but how accurate is this? In this work we use the 5 qubit code to answer this question.   

Quantum state and quantum entanglement protection using quantum measurements

The time evolution of some quantum states can be slowed down or even stopped under frequent measurements. This is the usual quantum Zeno effect. Here we report an operator quantum Zeno effect(Shu-Chao Wang, Ying Li, Xiang-Bin Wang,and Leong Chuan Kwek, PRL 110, 100505 (2013)), in which the evolution of some physical observables is slowed down through measurements even though thequantum state changes randomly with time. Based on the operator quantum Zeno effect, we show how we can protect quantum information from decoherence with two-qubit measurements, realizable with noisy two-qubit interactions. Besides, we report the quantum entanglement protection using weak measurement and measurement reversal scheme(Shu-Chao Wang, Zong-Wen Yu, Wen-Jie Zou, and Xiang-Bin Wang, PRA 89, 022318 (2014)). Exposed in the nonzero temperature environment, a quantum system can both lose and gain excitations by interacting with the environment. In this work, we show how to optimally protect quantum states and quantum entanglement in such a situation based on measurement reversal from weak measurement. In particular, we present explicit formulas of protection. We find that this scheme can circumvent the entanglement sudden death in certain conditions.   

Protecting a spin ensemble against decoherence in the strong-coupling regime of cavity QED

Hybrid quantum systems based on spin ensembles coupled to superconducting microwave cavities are promising candidates for robust experiments in cavity quantum electrodynamics (QED) and for future technologies employing quantum mechanical effects. We present recent experimental results of strong coupling between an ensemble of nitrogen-vacancy center electron spins in diamond and a superconducting microwave coplanar waveguide resonator. Although the coupling between a single spin and the electromagnetic field is typically rather weak, collective enhancement allows entering the strong coupling regime. Currently the main source of decoherence in these systems is inhomogeneous spin broadening, which limits their performance for the coherent transfer and storage of quantum information. Here we study the dynamics of a superconducting cavity strongly coupled to an ensemble of nitrogen-vacancy centers in diamond. We experimentally observe for the first time, how decoherence induced by inhomogeneous broadening can be suppressed in the strong-coupling regime, a phenomenon known as ``cavity protection" (Putz, S. et al. Nat. Phys. 10, 720--724 (2014)). To demonstrate the potential of this effect for coherent control schemes, we show how appropriately chosen microwave pulses can increase the amplitude of coherent oscillations between the cavity and spin ensemble by two orders of magnitude.   

Freely Scalable Quantum Technologies using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links

Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centres and in superconducting qubit clusters. We can regard such a system as a universal cell with diverse technological uses from communication to large-scale computing, provided that the cell is able to network with others and overcome any noise in the interlinks. We show that loss-tolerant entanglement purification makes quantum computing feasible with the noisy and lossy links that are realistic today: With a modestly complex cell design, and using a surface code protocol with a network noise threshold of $13.3\%$, we find that interlinks which attempt entanglement at a rate of $2$ MHz but suffer $98\%$ photon loss can result in kilohertz computer clock speeds (i.e. rate of high fidelity stabilizer measurements). Improved links would dramatically increase the clock speed. Our simulations employed local gates of a fidelity already achieved in ion trap devices.   

Resource requirements for a fault-tolerant quantum Fourier transform

The quantum Fourier transform (QFT) is a basic subroutine for most quantum algorithms providing an exponential speedup over classical ones. We investigate resource requirements for a fault-tolerant QFT. To implement single-qubit rotations for a QFT in a fault-tolerant manner, we examine three types of approaches: ancilla-free gate synthesis, ancilla-assisted gate synthesis, and state distillation. While the gate synthesis approximates single-qubit rotations with basic quantum operations, the state distillation enables to perform specific single-qubit rotations required for the QFT exactly. It is unknown, however, which approach is better for the QFT. We estimated the resource requirement for a QFT in each case, where the resource is measured by the total number of the $\pi/8$ gates denoted by $T$, which is called the $T$ count. Contrary to the initial expectation, the total $T$ count for the state distillation is considerably larger than those for the ancilla-free and ancilla-assisted gate synthesis. Thus, we conclude that the ancilla-assisted gate synthesis is the best for a fault-tolerant QFT so far.   

Quantum Fourier transform performance scaling; defective rotation gates

We investigate analytically and numerically the quantum Fourier transform (QFT) with defective controlled rotation (CROT) gates. We find that the QFT can tolerate systematic and random defects up to $30\%$ and still perform its function. Analytical scaling laws of QFT performance are derived with respect to the number of qubits $n$, the size $\delta$ of systematic defects, and size $\epsilon$ of random defects. Our analytical results are in excellent agreement with numerical simulations. In addition, we present an unexpected result: The performance of the defective QFT does not deteriorate with increasing $n$, but approaches a constant that scales in $\epsilon$. We derive an analytical formula that accurately reproduces the $\epsilon$ scaling of the performance plateaus. The extraordinary robustness of the QFT with respect to static gate defects displayed in our numerical and analytical calculations should be a welcome boon for laboratory and industrial realizations of quantum circuitry.   

Two results in topology, motivated by quantum computation

The field of quantum computation is built on the foundation of physics, mathematics, and computer science. While it has taken much from these fields, there are also interesting examples where it has given back. I will discuss two new results of this kind. In both cases, we use very basic ideas from quantum computation to prove an interesting fact about low-dimensional topology. First, we use the Solovay-Kitaev universality theorem with exponential precision to give a simple proof of the \#P-hardness of certain 3-manifold invariants. We then apply this result to show the existence of rather exotic 3-manifold diagrams. Second, we show a relationship between the distinguishing power of a link invariant, and the entangling power of the linear operator associated to braiding. More precisely, we show that link invariants derived from non-entangling solutions to the Yang-Baxter equation are trivial. The former is joint work with Catharine Lo (Caltech), and the latter is joint work with Stephen Jordan and Michael Jarett (UMD).   

Using Quantum Annealing Correction to Differentiate between Candidate Models

We study how the quantum annealing correction (QAC) strategy proposed by Pudenz et al.[1] allows us to distinguish between the final time statistics of the D-Wave device (DW2) and the classical rotor model (SSSV) proposed in ref.[2]. The SSSV model has been successful in reproducing the ground state probabilities for many Ising instances studied, setting a high bar for genuine quantum effects. Studying 1000 random instances using the QAC strategy with 112 logical qubits (448 physical qubits) [3], we show that the energy penalty term of the QAC strategy results in qualitatively different results for SSSV and DW2, with SSSV showing a clear separation in statistics for different penalty values while DW2 does not. While these results do not amount to a proof of quantumness, they support the notion that quantum effects play a relevant role in separating the SSSV model from the DW2 results observed. \\ \\ $[1]$ K. L. Pudenz, T. Albash, and D. A. Lidar, Nat Commun 5 (2014).\\ $[2]$ S. W. Shin, G. Smith, J. A. Smolin, and U. Vazirani, arXiv:1401.7087 (2014).\\ $[3]$ K. L. Pudenz, T. Albash, and D. A. Lidar, arXiv:1408.4382 (2014).