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Marmorino, M. G.

doi:10.1023/a:1021211206564

Cited by 7 publications

(6 citation statements)

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“…Starting from these inequalities and whithin this Gramian approach, a bounds for expectation values of quantum observables have been derived for pure states [12,[20][21][22][23][24][25][26] and in the context of classical methods for quantum chemistry. While certainly useful, this leaves a gap for these bounds to be applied in the NISQ era where noise is prevalent and quantum states and their evolutions are described by density operators.…”

Section: Review Of the Gramian Methodsmentioning

confidence: 99%

Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States

Weber¹,

Anand²,

Cervera-Lierta³

et al. 2021

Preprint

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Quantum computation in noisy, near-term implementations holds promise across multiple domains ranging from chemistry and many-body physics to machine learning, optimization, and finance. However, experimental and algorithmic shortcomings such as noise and decoherence lead to the preparation of imperfect states which differ from the ideal state and hence lead to faulty measurement outcomes, ultimately undermining the reliability of near-term quantum algorithms. It is thus crucial to accurately quantify and bound these errors via intervals which are guaranteed to contain the ideal measurement outcomes. To address this desire, based on the formulation as a semidefinite program, we derive such robustness intervals for the expectation values of quantum observables using only their first moment and the fidelity to the ideal state. In addition, to get tighter bounds, we take into account second moments and extend bounds for pure states based on the non-negativity of Gram matrices to mixed states, thus enabling their applicability in the NISQ era where noisy scenarios are prevalent. Finally, we demonstrate our results in the context of the variational quantum eigensolver (VQE) on noisy and noiseless simulations.

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Section: Review Of the Gramian Methodsmentioning

confidence: 99%

Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States

Weber¹,

Anand²,

Cervera-Lierta³

et al. 2021

Preprint

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“…The three fundamental inequalities governing this class are due to Temple, Weinstein, and Stevenson, all involving the computation of the mean value of the square of the Hamiltonian,

(〈〉, {\overset{H}{true}}^{2})

. An interesting review on these methods can be found in the work of Marmorino . The lower bound E W , to the true energy E 0 defined by the Weinstein inequality, e .…”

Section: Calculating Upper and Lower Bounds To The Energymentioning

confidence: 99%

“…An interesting review on these methods can be found in the work of Marmorino. 16 The lower bound E W , to the true energy E 0 defined by the Weinstein inequality, e.g., is:…”

Section: Calculating Upper and Lower Bounds To The Energymentioning

confidence: 99%

“…In the following, we use two types of trial functions: a 1s Slater orbital ϕ S = N S e −αr with N S = α 3=2 = ffiffiffi π p (Figure 1(b)) and an s-type Gaussian function ϕ SG = N SG e − αr 2 with N SG = (2α/π) 3/4 (Figure 1(a)]. As can be seen in Table 1, the analytical calculation of the integrals (16) and (17) without care results inVT 6 ¼TV [(S2.1) and (S2.2)], which is in total contradiction with the general result obtained in Eq. (11).…”

Section: Ementioning

confidence: 99%

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Calculating Hermitian Forms: The Importance of Considering Singular Points

Bhowmick

Hagebaum‐Reignier

Jeung

2019

Bulletin Korean Chem Soc

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In this paper, we point out the importance of boundary conditions in the evaluation of expectation values of quantum mechanical operators involved in upper bound (i.e., variational principle) or lower bound (e.g., methods using 〈〉trueH^bold2) calculations. The existence of singular points (or discontinuities) either in the trial function or in the operator itself needs to be carefully handled when calculating integrals, otherwise leads to non‐physical (e.g., imaginary) expectation values or to false values. In this case, the use of generalized functions (e.g., Heaviside or Dirac functions) is necessary to cure the singularity problems. As examples to put a stress on this mathematical subtleties, we discuss the wrong and true solutions obtained for the calculation of the mean values of the trueH^ and trueT^trueV^ (and trueV^trueT^) operators using two standard simple models: the one‐dimensional harmonic oscillator and the three‐dimensional hydrogen atom, along with Slater‐type and Gaussian trial functions.

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“…Recently, Taseli [22] studied the squared tangent potential V (x) = ν(ν − 1)tan 2 (x), x ∈ ( −π 2 , π 2 ) also known as the symmetric Pöschl-Teller potential [23,24]. It was noted that the energy eigenvalues, but not the eigenfunctions, of the Hamiltonian with the squared tangent potential on the symmetric interval ( −π 2 , π 2 ) are precisely the same as those of the Hamiltonian with the squared cotangent potential V (x) = ν(ν − 1)cot 2 (x) on the asymmetric interval (0, π) considered by Marmorino in [25] (see also [26]). This observation can be easily verified using the identity…”

Section: Trigonometric Tangent-squared Potentialmentioning

confidence: 99%

Darboux Transformation in the Tangent-Squared Potential and Supersymmetry

Sadeghi¹,

Pourdarvish

Rostami³

2014

ujpa

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In this paper, we consider one-dimensional Schrödinger equation for the trigonometric tangent-squared potential. Here, we construct the first-order Darboux transformation and the real valued condition of transformed potential for trigonometric tangent-squared potential equation. Also we obtain the transformed of potential and wave function. Finally, we discuss the correspondence between Darboux transformation and supersymmetry. In order to have supersymmetry and commutative and anti-commutative algebra, we obtain some condition for the corresponding equation.

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Cited by 7 publications

References 0 publications

Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States

Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States

Calculating Hermitian Forms: The Importance of Considering Singular Points

Darboux Transformation in the Tangent-Squared Potential and Supersymmetry

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