Session D39: Quantum Protocols: Tomography, Communication, and Coding

Practical characterization of quantum devices without tomography

Quantum tomography is the main method used to assess the quality of quantum information processing devices, but its complexity presents a major obstacle for the characterization of even moderately large systems. Part of the reason for this complexity is that tomography generates much more information than is usually sought. Taking a more targeted approach, we develop schemes that enable (i) estimating the ?delity of an experiment to a theoretical ideal description, (ii) learning which description within a reduced subset best matches the experimental data. Both these approaches yield a signi?cant reduction in resources compared to tomography. In particular, we show how to estimate the ?delity between a predicted pure state and an arbitrary experimental state using only a constant number of Pauli expectation values selected at random according to an importance-weighting rule. In addition, we propose methods for certifying quantum circuits and learning continuous-time quantum dynamics that are described by local Hamiltonians or Lindbladians.   

Quantum tomography with small number of copies: a simple estimator for qubit states

In quantum tomography, one performs repeated measurements on $N$ copies of a given but unknown state and constructs an estimator for the state from the gathered data. A common way of converting the data into an estimator is the maximum-likelihood (ML) method, where the estimator is the state with the largest probability of giving rise to the observed data. ML methods methods work well for large $N$, since the likelihood function for large $N$ is sharply peaked around its maximum. For small $N$, however, there is a significant neighborhood of states around the maximum with nearly equal probability of giving rise to the data. One can then imagine using the likelihood function as a weight to construct an estimator as an average over states. This motivates the introduction of the ``mean estimator,'' also previously discussed for quantum tomography in the spirit of Bayesian estimation by Blume-Kohout [NJP 12, 043034(2010)]. Here, we extend the mean estimator for a classical die problem to an estimator for qubit states, and demonstrate its advantage over ML estimators. We also discuss a way of overcoming the common complaint of rank-deficiency in ML estimators for our estimator. This simple estimator should be useful as a convenient first estimate for any qubit tomography experiment.   

Joint quantum tomography of state preparation and measurements using only known quantum operations

An important problem in quantum information is the complete characterization of quantum devices, which is usually referred to as ``quantum tomography''. Quantum tomography procedures exist for the characterization of quantum states (quantum state tomography), operations (quantum process tomography) and measurements (quantum measurement tomography) --- and each of these procedures can be performed using only product states, local unitary operations and local projective measurements. Here we consider the problem of jointly characterizing both the initial state as well as the measurement observable of a system using only well characterized quantum operations. We find that neither local unitaries nor local completely-positive trace-preserving maps are sufficient for obtaining a complete description of the state preparation and measurement of the system, and describe a scheme that uses non-unital trace-reducing physical maps to obtain such a description.   

Single shot quantum state estimation via continuous measurement in a strong back-action regime

Quantum state reconstruction is a fundamental task in quantum information science. The standard approach employs many projective measurements on a series of identically prepared systems in order to collect sufficient statistics of an informationally complete set of observables. An alternative procedure is to reconstruct quantum state by performing weak continuous measurement collectively on an ensemble, while simultaneously applying time varying controls [1]. For known dynamics, the measurement history determines the initial state. In current implementations the shot noise of the probe dominates over projection noise so that measurement-induced backaction is negligible. We generalize this to the regime where quantum backaction can play a significant role, even for small numbers of particles. Using the framework of quantum filtering theory, we model the reconstruction of the state of a qubit through collective spin measurement via the Faraday interaction and magnetic field controls, and develop a maximum-likelihood estimate based on the Fisher information contained in the measurement record. \\[4pt] [1] A. Silberfarb and I. H. Deutsch, ``Quantum-state reconstruction via continuous measurement,'' Phys. Rev. Lett. 95, 030402 (2005).   

Beyond maximum-likelihood estimation

When the estimators in quantum state tomography or quantum process tomography are obtained by maximizing the likelihood, which has become the method of choice, a unique result is not obtained if the data are informationally incomplete. By combining maximum-likelihood (ML) estimation with Jaynes's maximum-entropy (ME) principle, a unique estimator can be determined, and this is possible by an efficient iterative algorithm. The resulting estimators, however, can have the familiar deficiencies of maximum-likelihood estimators. Alternative estimation procedures that avoid these drawbacks are wanted. The talk reports on MLME estimation as well as alternative approaches with a Bayesian touch. [References: Phys. Rev. Lett. 107 (2011) 020404; arXiv:1110.1202]   

Experimental Realization of Adaptive Qubit Tomography

In quantum state tomography, an informationally complete set of measurements is made on N identically prepared quantum systems and from these measurements the quantum state can be determined. In the limit as $N \rightarrow \infty$, the estimation of the state converges on the true state. The rate at which this convergence occurs depends on both the state and the measurements used to probe the state. To characterize the quality of a set of measurements the fidelity of the estimation with the true state, averaged over a prior distribution of states, is used as a figure of merit. It is known [1] that for states very close to the surface of the bloch sphere, the average infidelity ($1-F$) goes down with a rate proportional to $\frac{1}{\sqrt{N}}$. It has also been shown that there exists a gap between collective measurement protocols and local measurement protocols, but that local \textit{adaptive} measurement protocols can come close to saturating the collective measurement bound of $\frac{1}{N}$ [2]. Here we present an experimental demonstration of one qubit tomography that achieves a rate of convergence of $\frac{1}{N}$ with only a single adaptive step and local measurements.\\[4pt] [1] Phys. Rev. A 78, 052122 (2008)\\[0pt] [2] Phys. Rev. Lett. 97, 130501 (2006)   

Scalable estimation of computational gate fidelities

Scalable methods for characterizing the noise affecting a quantum system are of significant interest in theoretical and experimental quantum information theory. Since completely characterizing the noise is exponentially hard in the number of qubits comprising the system, there has been significant effort in developing scalable methods for characterizing particular features of the noise. In particular, ``randomized benchmarking'' has been shown to be a robust and scalable method for estimating the average error rate over the set of quantum computational gates. We propose a new protocol that allows for benchmarking individual quantum gates rather than the average over the entire set. The protocol consists of a mixture of deterministic and random applications of computational gates and is shown to be scalable in the number of qubits comprising the system. The method is robust against state preparation and measurement errors and is valid provided the average variation of the noise over the gate set can be sufficiently bounded.   

Quantum controlled paths for perfect discrimination of no-signalling channels

A no-signalling channel transforming systems in Alice's and Bob's local laboratories is compatible with two different causal structures: one where Alice's output can be sent to Bob's input and another where Bob's output can be sent to Alice's input. I show that a quantum superposition of circuits operating within these two causal structures enables the perfect discrimination between two no-signalling channels that could not be perfectly distinguished by any ordinary circuit [1]. Such a quantum superposition can be in principle achieved by introducing a qubit that controls the path followed by quantum systems, routing them to different ports of the given no-signalling channel. \\[4pt] [1] G. Chiribella, \emph{Perfect discrimination of no-signalling channels via quantum superposition of causal structures}, arXiv:1109.5154.   

Experimental characterization of coherent dynamics in a spin chain

We experimentally characterize the coherent room-temperature magnetization dynamics of a spin chain evolving under an effective double-quantum Hamiltonian. Our results indicate that a localized magnetic moment travels down the chain with a group velocity of $6.04\pm0.38$ $\mu$m/s, corresponding to coherent transport over $N\approx 26$ spins on the timescale of the experiment. We also characterize the influence of the ends of the chains on the magnetization dynamics. Our results are in excellent agreement with a nearest-neighbor-coupled analytical model that predicts that the dynamics are restricted to a Liouville space that only grows quadratically with the number of spins. This suggests that the long-range couplings present in the experimental system only cause a slow leakage out of the subspace. As the double-quantum Hamiltonian is related to the standard one-dimensional XX Hamiltonian by a similarity transform, our results can be directly extended to XX quantum spin chains, which have been extensively studied in the context of both quantum magnetism and quantum information processing   

Experimental Monte Carlo Quantum Process Certification

Experimental implementations of quantum information processing have now reached a state, at which quantum process tomography starts to become impractical, since the number of experimental settings as well as the computational cost of the post processing required to extract the process matrix from the measurements scales exponentially with the number of qubits in the system. In order to determine the fidelity of an implemented process relative to the ideal one, a more practical approach called Monte Carlo quantum process certification was proposed in Ref.~[1]. Here we present an experimental implementation of this scheme in a circuit quantum electrodynamics setup. Our system is realized with three superconducting transmon qubits coupled to a coplanar microwave resonator which is used for the joint-readout of the qubit states. We demonstrate an implementation of Monte Carlo quantum process certification and determine the fidelity of different two- and three-qubit gates such as \textsc{cphase}-, \textsc{cnot}-, \textsc{2cphase}- and Toffoli-gates. The obtained results are compared with the values obtained from conventional process tomography and the errors of the obtained fidelities are determined. \\[4pt] [1] M.~P.~da~Silva, O.~Landon-Cardinal and D.~Poulin, arXiv:1104.3835(2011)   

Quantum filtering one bit at a time

We consider the purification of a quantum state using the information obtained from a continuous measurement record, where the classical measurement record is digitized to a single bit per measurement after the measurements have been made. Analysis indicates that efficient and reliable state purification is achievable for one- and two-qubit systems. We also consider quantum feedback control based on the discrete one-bit measurement sequences.   

New results in fault-tolerant quantum computing

We compare the accuracy of two methods used to construct a logical zero state appropriate for the [7,1,3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment. The first method is to apply error correction, via syndrome measurement, on seven physical qubits all in the state zero, using four-qubit Shor states to implement the syndrome measurements. The second method is to directly implement the [7,1,3] encoding gate sequence. We find surprising results that show that even at the most basic level there is still much to be learned about achieving fault tolerance.   

On the Use of Shor States for the [7,1,3] Quantum Error Correcting Code

We explore the effect of Shor state construction methods on logical state encoding and quantum error correction for the [7,1,3] Calderbank-Shor-Steane quantum error correction code in a nonequiprobable error environment. We determine the optimum number of verification steps to be used in Shor state construction and whether Shor states without verification are usable for practical quantum computation. These results are compared to the same processes of encoding and error correction where Shor states are not used. We demonstrate that the construction of logical zero states with no first order error terms may not require the complete edifice of quantum fault tolerance. With respect to error correction, we show for a particular initial state that error correction using a single qubit for syndrome measurement yields a similar output state accuracy to error correction using Shor states as syndrome qubits. In addition, we demonstrate that error correction with Shor states has an inherent sensitivity to bit-flip errors.   

Quantum codes with low weight stabilizers

We study quantum cyclic stabilizer codes whose stabilizer can be always defined by one or two stabilizer generators. Our main goal is to construct low-weight stabilizer generators that can yield quantum codes with high code rate and simple error correction. To do so, we apply the classical quaternary representation of stabilizer codes and extend our recent study of one-generator cyclic codes [1]. For any stabilizer generator of weight four or five, we formulate a necessary and sufficient condition for its commutativity. We then proceed with a design of additive cyclic codes with such generators. In some cases, we also extend our commutativity condition and code design to generators of weight six. In particular, quantum cyclic codes with stabilizers of weight four are mapped to the generalized toric codes. Here we also extend the notion of toric codes using a translationally invariant generator and periodic boundary conditions on a two dimensional lattice. Some of our numerically constructed codes can be redefined by means of Code Word Stabilized (CWS) representation [1] as quantum versions of repetition codes. We particularly concentrate on codes with a fixed nonzero rate for which the minimum distance asymptotically grows as the blocklength grows.\\[4pt] [1] arXiv:1108.5490v1