Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.
We discuss the state tomography of a fixed qubit (a spin-1/2 target particle), which is in general in a mixed state, through 1D scattering of a probe qubit off the target. Two strategies are presented, by making use of different degrees of freedom of the probe, spin and momentum. Remarkably, the spatial degree of freedom of the probe can be useful for the tomography of the qubit.
State tomography on qubit pairs is routinely carried out by measuring the two qubits separately, while one expects a higher efficiency from tomography with highly symmetric joint measurements of both qubits. Our numerical study of simulated experiments does not support such expectations.
We report on the implementation of quantum state tomography for an ensemble of Eu$^{3+}$ dopant ions in a \YSO crystal. The tomography was applied to a qubit based on one of the ion's optical transitions. The qubit was manipulated using optical pulses and measurements were made by observing the optical free induction in a phase sensitive manner. Fidelities of $>90$% for the combined preparation and measurement process were achieved. Interactions between the ions due to the change in the ions' permanent electric dipole moment when excited optically were also measured. In light of these results,…
The efficiency of quantum state tomography is discussed from the point of view of quantum parameter estimation theory, in which the trace of the weighted covariance is to be minimized. It is shown that tomography is optimal only when a special weight is adopted.
Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke st…
A new qubit tomography protocol is introduced, based on a continuous positive operator valued measure, which is supported by the set of pure states, and equivariant under the symmetry group SO(3,R) of the qubit state space. Thus the sample data for the tomography protocol is a discrete set of points on the Bloch sphere obtained by a series of independent measurements on identical replicas of the qubit. Although the maximum likelihood estimator uniquely exists, it cannot be explicitly calculated. Instead, we present another convenient, simple estimator, show that it is unbiased and calculate it…
We demonstrate high-fidelity Zeeman qubit state detection in a single trapped 88 Sr+ ion. Qubit readout is performed by shelving one of the qubit states to a metastable level using a narrow linewidth diode laser at 674 nm followed by state-selective fluorescence detection. The average fidelity reached for the readout of the qubit state is 0.9989(1). We then measure the fidelity of state tomography, averaged over all possible single-qubit states, which is 0.9979(2). We also fully characterize the detection process using quantum process tomography. This readout fidelity is compatible with recent…
We propose a method for precision statistical control of quantum processes based on superconductor phase qubits. Using the universal quantum tomography method, we provide a detailed analysis of accuracy of tomography for a 2-qubit gate SQiSW, which arises due to capacitive coupling between qubits. The developed approach could be successfully applied for quality and efficiency problems of superconductor quantum information technologies.
We present an example of quantum process tomography performed on a single solid state qubit. The qubit used is two energy levels of the triplet state in the Nitrogen-Vacancy defect in Diamond. Quantum process tomography is applied to a qubit which has been allowed to decohere for three different time periods. In each case the process is found in terms of the $χ$ matrix representation and the affine map representation. The discrepancy between experimentally estimated process and the closest physically valid process is noted.
The optimal state determination (or tomography) is studied for a composite system of two qubits when measurements can be performed on one of the qubits and interactions of the two qubits can be implemented. The goal is to minimize the number of interactions to be used. The algebraic method applied in the paper leads to an extension of the concept of mutually unbiased measurements.
Optimal generalized measurements for state estimation are well understood. However, practical quantum state tomography is typically performed using a fixed set of projective measurements and the question of how to choose these measurements has been largely unexplored in the literature. In this work we develop theoretical asymptotic bounds for the average fidelity of pure qubit tomography using measurement sets whose axes correspond to vertices of Platonic solids. We also present complete simulations of maximum likelihood tomography for mixed qubit states using the Platonic solid measurements. …
Tomography is an indispensable part of quantum computation as it enables diagnosis of a quantum process through state reconstruction. Existing tomographic protocols are based on determining expectation values of various Pauli operators which typically require single-qubit rotations. However, in realistic systems, qubits often develop some form of unavoidable stray coupling making it difficult to manipulate one qubit independent of its partners. Consequently, standard protocols applied to those systems result in unfaithful reproduction of the true quantum state. We have developed a protocol, ca…
In this paper, we derive analytic expressions for the starting (initial) values of the parameters of the T-matrix that is frequently employed in the construction of a theoretical density matrix in a Maximum Likelihood Estimate (MLE) procedure; this optimization procedure is used to fit the experimental data pertaining to single qubit state tomography. Appropriate starting values are critical in achieving a global minimum in the fitting process. We also indicate an analytic way of making the experimentally determined density matrix physical without resorting to the MLE process, if optimality in…
We review single-qubit quantum process tomography for trace-preserving and nontrace-preserving processes, and derive explicit forms of the general constraints for fitting experimental data. These forms provide additional insight into the structure of the process matrix as well as reveal a tighter bound on the trace of a nontrace-preserving process than has been previously stated. We also describe, for completeness, how to incorporate measured imperfect input states.
We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characteriz…
We present, and analyze thoroughly, a highly symmetric and efficient scheme for the determination of a single-qubit state, such as the polarization properties of photons emitted by a single-photon source. In our scheme there are only four measured probabilities, just enough for the determination of the three parameters that specify the qubit state, whereas the standard procedure would measure six probabilities.
We briefly review single-qubit quantum process tomography for trace-preserving and nontrace-preserving processes, and derive explicit forms of the general constraints for fitting experimental data. These new forms provide additional insight into the structure of the process matrix. We illustrate their utility with several examples, including a discussion of qubit leakage error models and the intuition which can be gained from their process matrices.
We used quantum process tomography to investigate and identify the function of a nonideal two-qubit quantum-state filters subject to various degree of decoherence. We present a simple decoherence model that explains the experimental results and point out that a beamsplitter followed by a post-selection process is not, as commonly believed, a singlet-state filter. In the ideal case it is a triplet-state filter.
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
We use quantum detector tomography to characterize the qubit readout in terms of measurement POVMs on IBM Quantum Computers IBM Q 5 Tenerife and IBM Q 5 Yorktown. Our results suggest that the characterized detector model deviates from the ideal projectors by a few percent. Further improvement on this characterization can be made by adopting two- or more-qubit detector models instead of independent single-qubit detectors for all the qubits in one device. An unexpected behavior was seen in the physical qubit labelled as qubit 3 of IBM Q 5 Tenerife, which can be a consequence of detector crosstal…
Full quantum state tomography is used to characterize the state of an ensemble based qubit implemented through two hyperfine levels in Pr3+ ions, doped into a Y2SiO5 crystal. We experimentally verify that single-qubit rotation errors due to inhomogeneities of the ensemble can be suppressed using the Roos-Moelmer dark state scheme. Fidelities above >90%, presumably limited by excited state decoherence, were achieved. Although not explicitly taken care of in the Roos-Moelmer scheme, it appears that also decoherence due to inhomogeneous broadening on the hyperfine transition is largely suppressed…
Quantum process tomography has become increasingly critical as the need grows for robust verification and validation of candidate quantum processors. Here, we present an approach for efficient quantum process tomography that uses a physically motivated ansatz for an unknown quantum process. Our ansatz bootstraps to an effective description for an unknown process on a multi-qubit processor from pairwise two-qubit tomographic data. Further, our approach can inherit insensitivity to system preparation and measurement error from the two-qubit tomography scheme. We benchmark our approach using nume…
We present an empirical strategy to determine the Hamiltonian dynamics of a two-qubit system using only initialization and measurement in a single fixed basis. Signal parameters are estimated from measurement data using Bayesian methods from which the underlying Hamiltonian is reconstructed, up to three unobservable phase factors. We extend the method to achieve full control Hamiltonian tomography for controllable systems via a multi-step approach. The technique is demonstrated and evaluated by analyzing data from simulated experiments including projection noise.
A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space $(-1,1)^3$ of the Stokes parameters $(ξ_1, ξ_2,ξ_3)$ that specify qubit states should be regarded as a Riemannian manifold endowed with a metric $g_{ij}:=δ_{ij}/(1-(ξ_i)^2)$, and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric $g_{ij}$. An efficient algorithm for computing the ma…
We demonstrate the generation of heralded Fock-basis qubits $(a|0\rangle + b|1\rangle)$ from transient collective spin excitations in a hot atomic vapor. The collective spin excitation is heralded by Raman-scattered photons in a four-wave mixing process seeded by a weak coherent optical excitation. The amplitude and phase of the seed field allow arbitrary control over the qubit coefficients. The state of the ensemble is read out optically and then characterized using balanced homodyne tomography. This work represents a step towards quantum engineering of collective atomic spin states.
We present an example of quantum process tomography (QPT) performed on a single solid state qubit. The qubit used is two energy levels of the triplet state in the Nitrogen-Vacancy defect in Diamond. Quantum process tomography is applied to a qubit which has been allowed to decohere for three different time periods. In each case the process is found in terms of the chi matrix representation and the affine map representation. The discrepancy between experimentally estimated process and the closest physically valid process is noted. The results of QPT performed after three different decoherence t…
The standard method of Quantum State Tomography (QST) relies on the measurement of a set of noncommuting observables, realized in a series of independent experiments. Ancilla Assisted QST (AAQST) proposed by Nieuwenhuizen and co-workers (Phys. Rev. Lett., 92, 120402 (2004)) greatly reduces the number of independent measurements by exploiting an ancilla register in a known initial state. In suitable conditions AAQST allows mapping out density matrix of an input register in a single experiment. Here we describe methods for explicit construction of AAQST experiments in multi-qubit registers. We a…
As the variety of commercially available quantum computers continues to increase so does the need for tools that can characterize, verify and validate these computers. This work explores using quantum state tomography for characterizing the performance of individual qubits and develops a vector field visualization for presentation of the results. The proposed protocol is demonstrated in simulation and on quantum computing hardware developed by IBM. The results identify qubit performance features that are not reflected in the standard models of this hardware, indicating opportunities to improve…
Dynamical decoupling is a technique that protects qubits against noise. The ability to preserve quantum coherence in the presence of noise is essential for the development of quantum devices. Here the Rigetti quantum computing platform was used to test different dynamical decoupling sequences. The performance of the sequences was characterized by Quantum Process Tomography and analyzed using the quantum channels formalism. It is shown that dynamical decoupling can reduce qubit dephasing but cannot protect against spontaneous emission. Furthermore, from process tomography results, it was also p…
Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.
We have investigated the realizability of the controlled-NOT (CNOT) gate and characterized the gate operation by quantum process tomography for a chain of qubits, realized by electrons confined in self-assembled quantum dots embedded in the spin field-effect transistor. We have shown that the CNOT gate operation and its process tomography are performable by using the spin exchange interaction and several local qubit rotations within the coherence time of qubits. Moreover it is shown that when the fluctuation of operation time and the imperfection of polarization of channel electrons are consid…
The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal. The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single pl…
A single photon, delocalized over two optical modes, is characterized by means of quantum homodyne tomography. The reconstructed four-dimensional density matrix extends over the entire Hilbert space and thus reveals, for the first time, complete information about the dual-rail optical quantum bit as a state of the electromagnetic field. The experimental data violate the Bell inequality albeit with a loophole similar to the detection loophole in photon counting experiments.
In circuit-based quantum computing, the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits. In certain physical architectures, however, some qubits may be 'hidden' and lacking direct addressability through dedicated control and readout lines, for instance because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems. In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measu…
We have implemented a universal quantum logic gate between qubits stored in the spin state of a pair of trapped calcium 40 ions. An initial product state was driven to a maximally entangled state deterministically, with 83% fidelity. We present a general approach to quantum state tomography which achieves good robustness to experimental noise and drift, and use it to measure the spin state of the ions. We find the entanglement of formation is 0.54.
Tomography of the two qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the Quantum Key Distribution (QKD) scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so called SIC POVM qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimen…
We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. We focus on two apparently simple cases, full tomography of one and two qubit systems. We analyze and compare those techniques according to two figures of merit. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic pro…
We experimentally demonstrate that loop state-preparation-and-measurement (SPAM) tomography is capable of detecting correlated errors in a two-qubit system. We prepare photon pairs in a state that approximates a Werner state, which may or may not be entangled. By performing measurements with multiple different detector settings we are able to detect correlated errors between two single-qubit measurements performed in different locations. No assumptions are made concerning either the state preparations or the measurements, other than that the dimensions of the states and the positive-operator-v…
Benchmarking and characterising quantum states and logic gates is essential in the development of devices for quantum computing. We introduce a Bayesian approach to self-consistent process tomography, called fast Bayesian tomography (FBT), and experimentally demonstrate its performance in characterising a two-qubit gate set on a silicon-based spin qubit device. FBT is built on an adaptive self-consistent linearisation that is robust to model approximation errors. Our method offers several advantages over other self-consistent tomographic methods. Most notably, FBT can leverage prior informatio…
The success of quantum information processing applications relies on accurate and efficient characterization of quantum states, especially nearly-pure states. In this work, we investigate a procedure for adaptive qubit state tomography which achieves $O(1/N)$ scaling in accuracy for large $N$. We analyze the performance of the adaptive protocol on the characterization of pure, mixed and nearly-pure states, and clarify the interplay between the purity of the state and the statistical fluctuations of the measurement results. Our results highlight the subtle difference between the characterizatio…
Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exist, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.
Quantum logic gates must perform properly when operating on their standard input basis states, as well as when operating on complex superpositions of these states. Experiments using superconducting qubits have validated the truth table for particular implementations of e.g. the controlled-NOT gate [1,2], but have not fully characterized gate operation for arbitrary superpositions of input states. Here we demonstrate the use of quantum process tomography (QPT) [3,4] to fully characterize the performance of a universal entangling gate between two superconducting quantum bits. Process tomography …
Adaptive techniques have important potential for wide applications in enhancing precision of quantum parameter estimation. We present a recursively adaptive quantum state tomography (RAQST) protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this RAQST protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our RAQST protocol can outperform the tomography protocols using mutually unbiased bases (MUB) and the two-stage MUB adapti…
We investigate the performance of a Kennedy receiver, which is known as a beneficial tool in optical coherent communications, to the quantum state discrimination of the two superpositions of vacuum and single photon states corresponding to the $\hatσ_x$ eigenstates in the single-rail encoding of photonic qubits. We experimentally characterize the Kennedy receiver in vacuum-single photon two-dimensional space using quantum detector tomography and evaluate the achievable discrimination error probability from the reconstructed measurement operators. We furthermore derive the minimum error rate ob…
Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pul…
We propose and analyze quantum state estimation (tomography) using continuous quantum measurements with resource limitations, allowing the global state of many qubits to be constructed from only measuring a few. We give a proof-of-principle investigation demonstrating successful tomographic reconstruction of an arbitrary initial quantum state for three different situations: single qubit, remote qubit, and two interacting qubits. The tomographic reconstruction utilizes only a continuous weak probe of a single qubit observable, a fixed coupling Hamiltonian, together with single-qubit controls. I…
We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The ba…
The precision limit in quantum state tomography is of great interest not only to practical applications but also to foundational studies. However, little is known about this subject in the multiparameter setting even theoretically due to the subtle information tradeoff among incompatible observables. In the case of a qubit, the theoretic precision limit was determined by Hayashi as well as Gill and Massar, but attaining the precision limit in experiments has remained a challenging task. Here we report the first experiment which achieves this precision limit in adaptive quantum state tomography…
Time-bin qubits, where information is encoded in a single photon at different times, have been widely used in optical fiber and waveguide based quantum communications. With the recent developments in distributed quantum computation, it is logical to ask whether time-bin encoded qubits may be useful in that context. We have recently realized a time-bin qubit controlled-phase (C-Phase) gate using a 2 X 2 optical switch based on a lithium niobate waveguide, with which we demonstrated the generation of an entangled state. However, the experiment was performed with only a pair of input states, and …
In this work we present an algorithm of building an adequate model of polarizing quantum state measurement. This model takes into account chromatic aberration of the basis change transformation caused by the parasitic dispersion of the wave plates crystal and finite radiation spectral bandwidth. We show that the chromatic aberration reduces the amount of information in the measurements results. Using the information matrix approach we estimate the impact of this effect on the qubit state reconstruction fidelity for different values of sample size and spectral bandwidth. We also demonstrate tha…
