Minimax quantum state estimation under Bregman divergence

Quadeer, Maria; Tomamichel, Marco; Ferrie, Christopher

doi:10.22331/q-2019-03-04-126

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…Bayesian inference ran to completion 8 for of the gates in the range 4 • 10 −5 up to 2 • 10 −4 . For these 8 We note that the larger values of for which inference did not complete in this case may still yield results if the number of particles is increased, given that the noisy super-operators obtained with smaller will still be in the support of the prior. This highlights the trade-offs one can explore between time, computational resources, and the strength of our assumptions about the buttons.…”

Section: Long-sequence Gate Set Tomographymentioning

confidence: 82%

“…Bayesian inference ran to completion 8 for of the gates in the range 4 • 10 −5 up to 2 • 10 −4 . For these values, inference was successful over essentially the full range of SPAM values.…”

Section: Button Label Prior Example Valuesmentioning

confidence: 99%

“…The problem then becomes determining how should we best learn Λ given experimental evidence from our quantum device. The tomography problem has been approached in a wide variety of ways [2][3][4][5][6][7][8][9][10][11][12]. The various procedures generally learn Λ by (repeatedly) preparing a variety of input states {ρ i }, sending each through the application of U, and then measuring a variety of effects {E j }.…”

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confidence: 99%

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Operational, gauge-free quantum tomography

Matteo

Gamble

Granade

et al. 2020

Quantum

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As increasingly impressive quantum information processors are realized in laboratories around the world, robust and reliable characterization of these devices is now more urgent than ever. These diagnostics can take many forms, but one of the most popular categories is tomography, where an underlying parameterized model is proposed for a device and inferred by experiments. Here, we introduce and implement efficient operational tomography, which uses experimental observables as these model parameters. This addresses a problem of ambiguity in representation that arises in current tomographic approaches (the gauge problem). Solving the gauge problem enables us to efficiently implement operational tomography in a Bayesian framework computationally, and hence gives us a natural way to include prior information and discuss uncertainty in fit parameters. We demonstrate this new tomography in a variety of different experimentally-relevant scenarios, including standard process tomography, Ramsey interferometry, randomized benchmarking, and gate set tomography.

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Section: Long-sequence Gate Set Tomographymentioning

confidence: 82%

Section: Button Label Prior Example Valuesmentioning

confidence: 99%

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confidence: 99%

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Operational, gauge-free quantum tomography

Matteo

Gamble

Granade

et al. 2020

Quantum

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“…The data hiding protocol can be implemented by sampling randomly from the full unitary group but it is sufficient to sample randomly from a unitary 2-design. Other applications of unitary 2-designs are the fidelity estimation of quantum channels [18], quantum state and process tomography [19], and more recently minimax quantum state estimation [20]. In quantum information theory, they have been used extensively in the analysis of decoupling [13], [15], [21], [22].…”

Section: Introductionmentioning

confidence: 99%

Kerdock Codes Determine Unitary 2-Designs

Can

Rengaswamy

Calderbank

et al. 2020

IEEE Trans. Inform. Theory

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The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length N = 2 m over Z4. We show that exponentiating these Z4-valued codewords by ı √ −1 produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form N + 1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL(2, N). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.

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Section: Introductionmentioning

confidence: 99%

Kerdock Codes Determine Unitary 2-Designs

Can

Rengaswamy

Calderbank

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length N = 2 m over Z4. We show that exponentiating these Z4-valued codewords by ı √ −1 produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form N + 1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL(2, N ). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.

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Minimax quantum state estimation under Bregman divergence

Cited by 6 publications

References 26 publications

Operational, gauge-free quantum tomography

Operational, gauge-free quantum tomography

Kerdock Codes Determine Unitary 2-Designs

Kerdock Codes Determine Unitary 2-Designs

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