Pauli Basis Research Articles

Efficient error models for fault-tolerant architectures and the Pauli twirling approximation

The design and optimization of realistic architectures for fault-tolerant quantum computation requires error models that are both reliable and amenable to large-scale classical simulation. Perhaps the simplest and most practical general-purpose method for constructing such an error model is to twirl a given completely positive channel over the Pauli basis, a procedure we refer to as the Pauli twirling approximation (PTA). In this work we test the accuracy of the PTA for a small stabilizer measurement circuit relevant to fault-tolerant quantum computation, in the presence of both intrinsic gate errors and decoherence, and find excellent agreement over a wide range of physical error rates. The combined simplicity and accuracy of the PTA, along with its direct connection to the chi matrix of process tomography, suggests that it be used as a standard reference point for more refined error model constructions.

Complete positivity of the map from a basis to its dual basis

The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of Mn to its dual basis is a complete order isomorphism. We exhibit “natural” orthonormal bases for Mn such that this map is an order isomorphism, but not a complete order isomorphism. Included among such bases is the Pauli basis. Our results generalize the Choi matrix by giving conditions under which the role of the standard basis {Eij} can be replaced by other bases.

The robustness of magic state distillation against errors in Clifford gates

Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state~\cite{Bravyi}. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.

Universal Quantum Gate Set Approaching Fault-Tolerant Thresholds with Superconducting Qubits

We use quantum process tomography to characterize a full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits. All extracted gate fidelities, including those for Clifford group generators, single-qubit π/4 and π/8 rotations, and a two-qubit controlled-not, exceed 95% (98%), without (with) subtracting state preparation and measurement errors. Furthermore, we introduce a process map representation in the Pauli basis which is visually efficient and informative. This high-fidelity gate set serves as a critical building block towards scalable architectures of superconducting qubits for error correction schemes and pushes up on the known limits of quantum gate characterization.

Algorithmic approach to simulate Hamiltonian dynamics and an NMR simulation of quantum state transfer

We propose an iterative algorithm to simulate the dynamics generated by any $n$-qubit Hamiltonian. The simulation entails decomposing the unitary time evolution operator $U$ (unitary) into a product of different time-step unitaries. The algorithm product-decomposes $U$ in a chosen operator basis by identifying a certain symmetry of $U$ that is intimately related to the number of gates in the decomposition. We illustrate the algorithm by first obtaining a polynomial decomposition in the Pauli basis of the $n$-qubit Quantum State Transfer unitary by Di Franco et. al. (Phys. Rev. Lett. 101, 230502 (2008)) that transports quantum information from one end of a spin chain to the other; and then implement it in Nuclear Magnetic Resonance to demonstrate that the decomposition is experimentally viable and well-scaled. We furthur experimentally test the resilience of the state transfer to static errors in the coupling parameters of the simulated Hamiltonian. This is done by decomposing and simulating the corresponding imperfect unitaries.

Upper bounds on the noise threshold for fault-tolerant quantum computing

We prove new upper bounds on the tolerable level of noise in a quantum circuit. Weconsider circuits consisting of unitary k-qubit gates each of whose input wires is subject todepolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentiallynoise-free. We assume that the output of the circuit is the result of measuring somedesignated qubit in the final state. Our main result is that for p > 1 - Θ(1/√k), theoutput of any such circuit of large enough depth is essentially independent of its input,thereby making the circuit useless. For the important special case of k = 2, our bound isp > 35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubitCNOT gate, then our bound becomes 29.3%. These bounds on p are numerically betterthan previous bounds, yet are incomparable because of the somewhat different circuitmodel that we are using. Our main technique is the use of a Pauli basis decomposition,in which the effects of depolarizing noise are very easy to describe.

Upper bounds on the noise threshold for fault-tolerant quantum computing

We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary $k$-qubit gates each of whose input wires is subject to depolarizing noise of strength $p$, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for $p>1-\Theta(1/\sqrt{k})$, the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of $k=2$, our bound is $p>35.7\%$. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes $29.3\%$. These bounds on $p$ are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.

Geometry of generalized depolarizing channels

A generalized depolarizing channel acts on an $N$-dimensional quantum system to compress the ``Bloch ball'' in ${N}^{2}\ensuremath{-}1$ directions; it has a corresponding compression vector. We investigate the geometry of these compression vectors and prove a conjecture of Dixit and Sudarshan [Phys. Rev. A 78, 032308 (2008)], namely, that when $N={2}^{d}$ (i.e., the system consists of $d$ qubits), and we work in the Pauli basis then the set of all compression vectors forms a simplex. We extend this result by investigating the geometry in other bases; in particular we find precisely when the set of all compression vectors forms a simplex.

Single-Pass Polarimetric SAR Interferometry for Vessel Classification

This paper presents a novel method for vessel classification based on single-pass polarimetric synthetic aperture radar (SAR) interferometry. It has been developed according to recent ship scattering studies that show that the polarimetric response of many types of vessels can be described by trihedral- and dihedral-like mechanisms. The adopted methodology is quite simple. The input interferometric data are decomposed in terms of the Pauli basis, and hence, one height image is derived for each simple mechanism. Then, the local maxima of these images are isolated, and a 3-D map of scatters is generated. The correlation of this map with the scattering distribution expected for a set of reference ships provides the final classification decision. The performance of the proposed method has been tested with the orbital SAR simulator developed at Universitat Politecnica de Catalunya. Different vessel models have been processed with a sensor configuration similar to the incoming TanDEM-X system. The analysis of diverse vessel bearings, vessel speeds, and sea states shows that the map of scatters matches reasonably the geometry of ships allowing a correct identification even for adverse environmental conditions.

Target Scattering Decomposition in Terms of Roll-Invariant Target Parameters

The Kennaugh-Huynen scattering matrix con-diagonalization is projected into the Pauli basis to derive a new scattering vector model for the representation of coherent target scattering. This model permits a polarization basis invariant representation of coherent target scattering in terms of five independent target parameters, the magnitude and phase of the symmetric scattering type introduced in this paper, and the maximum polarization parameters (orientation, helicity, and maximum return). The new scattering vector model served for the assessment of the Cloude-Pottier incoherent target decomposition. Whereas the Cloude-Pottier scattering type alpha and entropy H are roll invariant, beta and the so-called target-phase parameters do depend on the target orientation angle for asymmetric scattering. The scattering vector model is then used as the basis for the development of new coherent and incoherent target decompositions in terms of unique and roll-invariant target parameters. It is shown that both the phase and magnitude of the symmetric scattering type should be used for an unambiguous description of symmetric target scattering. Target helicity is required for the assessment of the symmetry-asymmetry nature of target scattering. The symmetric scattering type phase is shown to be very promising for wetland classification in particular, using polarimetric Convair-580 synthetic aperture radar data collected over the Ramsar Mer Bleue wetland site to the east of Ottawa, Ontario, Canada

Indoor C-band polarimetric interferometry observations of a mature wheat canopy

We present results from experiments carried out in the ground-based synthetic aperture radar (GB-SAR) facility at the University of Sheffield to ascertain the role of polarimetric interferometry in crop height retrieval. To this end, a mature wheat canopy, grown in outdoor conditions, was reassembled inside the GB-SAR chamber and imaged at C-band using a two-dimensional scan. This allowed fully polarimetric tomography and interferometry. Interferometry using the VV, HH, and VH polarization states shows that the HH and VH interferograms retrieve a height close to the top of the soil layer for all angles of incidence considered, whereas the height retrieved from the VV interferogram increases with angle of incidence. The use of a Pauli basis gives poor results, due to the different location of the scattering phase centers in the VV and HH channels. The use of arbitrary polarization states shows that the top of the soil can be very accurately estimated using left-circular polarization, whereas, for angles of incidence close to 45/spl deg/, a polarization state similar to VV can be used to retrieve the top of the canopy; hence crop height can be recovered as the difference of these two interferometric heights. Polarimetric coherence optimization techniques are also studied. Unconstrained coherence optimization gives very unstable results, due to the small number of available samples. Constrained optimization results in stable retrieved heights, and the retrieved polarization states agree well with the polarization synthesis results.

Indoor wide-band polarimetric measurements on maize plants: a study of the differential extinction coefficient

A study of the wide-band polarimetric backscatter of maize plants, measured in laboratory conditions, is presented. The backscatter slant-range profiles in both linear (H-V) and Pauli basis manifest a higher extinction coefficient in the vertical channel due to the dominant vertical orientation of the structure of corn plants. The difference between the horizontal and vertical range profiles as the wave penetrates into the volume is employed to retrieve the differential extinction coefficient. In addition, the polarimetric target decomposition, as proposed by Cloude and Pottier, is used to obtain range profiles of alpha, entropy, and anisotropy. All these results reveal important features to be accounted for in the retrieval of the biophysical parameters of agricultural crops by means of polarimetric synthetic aperture radar interferometry.

Derivation of Dirac's Equation from the Evans Wave Equation

The Evans wave equation [1] of general relativity is expressed in spinor form, thus producing the Dirac equation in general relativity. The Dirac equation in special relativity is recovered in the limit of Euclidean or flat spacetime. By deriving the Dirac equation from the Evans equation it is demonstrated that the former originates in a novel metric compatibility condition, a geometrical constraint on the metric vector qμused to define the Einstein metric tensor. Contrary to some claims by Ryder, it is shown that the Dirac equation cannot be deduced unequivocally from a Lorentz boost in special relativity. It is shown that the usually accepted method in Clifford algebra and special relativity of equating the outer product of two Pauli spinors to a three-vector in the Pauli basis leads to the paradoxical result X = Y = Z = 0. The method devised in this paper for deriving the Dirac equation from the Evans equation does not use this paradoxical result.

Beyond stabilizer codes .I. Nice error bases

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.

Approximate quantum error correction can lead to better codes

We present relaxed criteria for quantum error correction that are useful when the specific dominant quantum noise process is known. As an example, we provide a four-bit code that corrects for a single amplitude damping error. This code violates the usual Hamming bound calculated for a Pauli description of the error process and has no simple explanation in terms of the usual Pauli basis GF(4) codes.