Quantum inference of states and processes

Ježek, Miroslav; Fiurášek, Jaromı́r; Hradil, Z.

doi:10.1103/physreva.68.012305

Cited by 173 publications

(140 citation statements)

References 50 publications

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“…There is a huge literature regarding analysis quantum estimation errors or quantum statistics [52,53,54,55,56,57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82]. Our aim here is to give a very brief discussion of estimation errors in different QPT schemes through a special example.…”

Section: Finite Ensemble-size Effectsmentioning

confidence: 99%

“…The problem with ill-conditioning due to inversion is a of a different nature: it is a numerical error that leads to a nonpositive or non-completely-positive map. This problem, to a large extent, can be addressed by supplementary data analysis methods, such as ML estimation [25,52,53,54,55,56,57], Bayesian state estimation [58,59,60], and other reliable regularization or reconstruction methods [61,62,63]. In principle, all known QPT schemes (including DCQD) can be optimized by utilizing such statistical error reduction techniques.…”

Section: The Role Of Inversionmentioning

confidence: 99%

“…In particular, in the SQPT and AAPT schemes an inversion on experimental data is inevitable. This inversion may induce an ill-conditioning feature [25,52], i.e., small errors in experimental outcomes may give rise to large errors in the estimation of E, and can sometimes result in non-positive maps. It should be stressed that quantum dynamics obtained via the usual prescription of unitary evolution followed by a partial trace over the bath, is always positive when the initial state is a valid density matrix.…”

Section: The Role Of Inversionmentioning

confidence: 99%

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Quantum-process tomography: Resource analysis of different strategies

2008

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Characterization of quantum dynamics is a fundamental problem in quantum physics and quantum information science. Several methods are known which achieve this goal, namely Standard Quantum Process Tomography (SQPT), Ancilla-Assisted Process Tomography (AAPT), and the recently proposed scheme of Direct Characterization of Quantum Dynamics (DCQD). Here, we review these schemes and analyze them with respect to some of the physical resources they require. Although a reliable figure-of-merit for process characterization is not yet available, our analysis can provide a benchmark which is necessary for choosing the scheme that is the most appropriate in a given situation, with given resources. As a result, we conclude that for quantum systems where two-body interactions are not naturally available, SQPT is the most efficient scheme. However, for quantum systems with controllable two-body interactions, the DCQD scheme is more efficient than other known QPT schemes in terms of the total number of required elementary quantum operations.

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Section: Finite Ensemble-size Effectsmentioning

confidence: 99%

Section: The Role Of Inversionmentioning

confidence: 99%

Section: The Role Of Inversionmentioning

confidence: 99%

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Quantum-process tomography: Resource analysis of different strategies

2008

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“…The actual implementation can be benchmarked with the aid of quantum state and process tomography [12,16]. We use a maximum likelihood algorithm to reconstruct the density matrix and perform a non-parametric bootstrap for statistical error analysis [18]. Because the error correction protocol acts as a single qubit quantum channel, it can be characterized by a quantum process tomography on the system qubit (indicated as ρ sys in Fig 1(a)).…”

Section: Fig 1 (A)mentioning

confidence: 99%

Undoing a Quantum Measurement

et al. 2013

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In general, a quantum measurement yields an undetermined answer and alters the system to be consistent with the measurement result. This process maps multiple initial states into a single state and thus cannot be reversed. This has important implications in quantum information processing, where errors can be interpreted as measurements. Therefore, it seems that it is impossible to correct errors in a quantum information processor, but protocols exist that are capable of eliminating them if they affect only part of the system. In this work we present the deterministic reversal of a fully projective measurement on a single particle, enabled by a quantum error-correction protocol that distributes the information over three particles.Measurements on a quantum system irreversibly project the system onto a measurement eigenstate regardless of the state of the system. Copying an unknown quantum state is thus impossible because learning about a state without destroying it is prohibited by the nocloning theorem [1]. At first, this seems to be a roadblock for correcting errors in quantum information processors. However, the quantum information can be encoded redundantly in multiple particles and subsequently used by quantum error correction (QEC) techniques [2][3][4][5][6][7]. When one interprets errors as measurements, it becomes clear that such protocols are able to reverse a partial measurement on the system. In experimental realizations of error correction procedures, the effect of the measurement is implicitly reversed but its outcome remains unknown. Previous realizations of measurement reversal with known outcomes have been performed in the context of weak measurements where the measurement and its reversal are probabilistic processes [8][9][10][11]. We will show that it is possible to deterministically reverse measurements on a single particle.We consider a system of three two-level atoms where each can be described as a qubit with the basis states |0 , |1 . An arbitrary pure single-qubit quantum state is given by |ψ = α|0 + β|1 with |α| 2 + |β| 2 = 1 and α, β ∈ C. In the used error-correction protocol, the information of a single (system) qubit is distributed over three qubits by storing the information redundantly in the state α|000 + β|111 . This encoding is able to correct a single bit-flip by performing a majority vote and is known as the repetition code [12].A measurement in the computational basis states |0 , |1 causes a projection onto the σ z axis of the Bloch sphere and can be interpreted as an incoherent phase flip. Thus, any protocol correcting against phase-flips is sufficient to reverse measurements in the computational basis. The repetition code can be modified to protect against such phase-flip errors by a simple basis change from |0 , |1 to |± = 1/ √ 2(|0 ± |1 ). After this basis change each individual qubit is in an equal superposition of |0 and |1 and therefore it is impossible to gain any information about the encoded quantum information by measuring a single qubit along σ z . Because the repet...

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“…From this set of data the process matrix χ can be obtained by inverting the relation in (3). However, to avoid unphysical results caused by quantum noise in the measurement process, we employ an iterative maximum likelihood algorithm [20] in order to find the physical process E which most likely generated the measured data set. We choose the products of the single qubit states |ψ 1 = |S , |ψ 2 = |D , |ψ 3 = (|D +i|S )/ √ 2 and |ψ 4 = (|D +|S )/ √ 2 as the 16 input states necessary for a tomography of our two qubit quantum gates.…”

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confidence: 99%

Process Tomography of Ion Trap Quantum Gates

et al. 2006

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A crucial building block for quantum information processing with trapped ions is a controlled-NOT quantum gate. In this paper, two different sequences of laser pulses implementing such a gate operation are analyzed using quantum process tomography. Fidelities of up to 92.6(6) % are achieved for single gate operations and up to 83.4(8) % for two concatenated gate operations. By process tomography we assess the performance of the gates for different experimental realizations and demonstrate the advantage of amplitude-shaped laser pulses over simple square pulses. We also investigate whether the performance of concatenated gates can be inferred from the analysis of the single gates.PACS numbers: 03.65. Wj, 03.67.Lx, 32.80.Qk Processing information with well-controlled quantum systems has the fascinating perspective of being much more powerful than classical computers for certain applications. A promising candidate for the experimental realization of quantum computing are strings of ions stored in linear Paul traps [1] as recently demonstrated by various key experiments, including the preparation of multi-particle entangled states [2,3], quantum teleportation [4,5] and quantum error correction [6]. Quantum information processing depends on the ability to implement single qubit rotations and most importantly an entangling two-qubit quantum gate [7,8,9,10]. Proper characterization and understanding of the action of gate operations and their imperfections is of vital importance in order to successfully apply them in complex computations.Generally, the implementations of quantum gates are imperfect due to decoherence and various systematic error sources present in experimental setups. A proper description of such an operation which accounts for the possibly non-unitary evolution of the qubits is provided by quantum process tomography [11,12]. Process tomography has already been applied for characterizing quantum gates in NMR and linear-optics quantum computing [13,14,15]. Here, we show that process tomography is a valuable tool for comparing different ion trap quantum gate implementations and optimizing the experimental parameters. This way, we were able to improve our controlled-NOT (CNOT) gate fidelity from 71 % [7] to almost 93 %. Moreover, the action of two successively applied gate operations is investigated and compared to the predictions from the single gate tomography result.We realize entangling gates between 40 Ca + ions held in a linear trap [16]. Quantum information is stored in superpositions of the |S ≡ S 1/2 (m = −1/2) ground state and the metastable |D ≡ D 5/2 (m = −1/2) state and is manipulated by laser pulses at a wavelength of 729 nm exciting the electric quadrupole transition between those states. A focus size smaller than the inter-ion distance and precise control of the focus position allows us to address single qubits. Detection of the qubit's quantum state is achieved by scattering light on the S 1/2 ↔ P 1/2 dipole transition and detecting the presence or absence of resonance fluorescence of t...

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Quantum inference of states and processes

Cited by 173 publications

References 50 publications

Quantum-process tomography: Resource analysis of different strategies

Quantum-process tomography: Resource analysis of different strategies

Undoing a Quantum Measurement

Process Tomography of Ion Trap Quantum Gates

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