This time, we implemented statistic noise, a legacy in Qiskit_Aqua[1]. Blue Cat implements a state vector output. In actual quantum computation, the probability distribution of the existence of a solution must be output for a finite number of observations. The state vector output can be interpreted as the result of increasing the number of observations to the limit of infinity. Therefore, even if it is not a real calculation, the difference in the existence probability distributions of those solutions will appear as their uncertain fluctuations. This is shot noise. This noise is almost negligible if enough observations are made, but in real-world calculations, it is kept to a small number of observations so as not to increase the calculation time. Therefore, this noise is a powerful tool to simulate the calculation on the actual machine. If the uncertainty fluctuation of the j-th element of the Hamiltonian ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mo>∑</mo><mi>j</mi></msub><msub><mi>c</mi><mi>j</mi></msub><msub><mi>H</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">H=\sum_jc_jH_j</annotation></semantics></math>H=∑j​cj​Hj​﻿ is　﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>H</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\Delta H_j</annotation></semantics></math>ΔHj​﻿ , then this noise is ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>H</mi><mi>j</mi></msub><mo>=</mo><msub><mi>H</mi><mi>j</mi></msub><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>H</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\Delta H_j=H_j-\langle H_j \rangle</annotation></semantics></math>ΔHj​=Hj​−⟨Hj​⟩﻿ which becomes where , ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>H</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle H_j \rangle</annotation></semantics></math>⟨Hj​⟩﻿ is the expectation value taken from the state vector. From here, the shot noise is ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msub><mo>∑</mo><mi>j</mi></msub><msubsup><mi>c</mi><mi>j</mi><mn>2</mn></msubsup><mo stretchy="false">⟨</mo><mi mathvariant="normal">Φ</mi><mo>∣</mo><mi mathvariant="normal">Δ</mi><msub><mi>H</mi><mi>j</mi></msub><mo>∣</mo><mi mathvariant="normal">Φ</mi><msup><mo stretchy="false">⟩</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">S(H)=\sqrt{\sum_jc_j^2\langle \Phi \mid \Delta H_j \mid \Phi \rangle^2}</annotation></semantics></math>S(H)=∑j​cj2​⟨Φ∣ΔHj​∣Φ⟩2<svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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M1001 80h400000v40h-400000z"></path></svg>​﻿ The result is In this study, the ground state energy of molecular hydrogen was calculated for the number of shots = 10, 50, 100, and 1000 by varying the distance between hydrogen atoms from 0.1 to 2.5 (Å) in increments of 0.1. The depths of both the Hamiltonian and the clusters were set to 2, the UCCSD was used for the clusters, and the BFGS method was used for the optimization. Figure 1 shows that there is no significant difference in accuracy up to an inter-atomic distance of 1.5, but there is a slight difference from there. After that, the values are random and low precision values are occasionally found regardless of the number of shots. This is probably because the state vector in molecular hydrogen has many states that are not zero but have a slight probability of existence, and these states come out and become larger than the value of the state vector when the number of shots is increased. <img src="https://assets.blueqat.com/public/uploads/us-east-2:62eba507-efdc-48b7-85a9-2f09d417847c/2021/07/22/blogSN1.png"/>Figure 1: Ground energy levels with shot noise when the number of shots is set to 10, 50, 100, and 1000. <img src="https://assets.blueqat.com/public/uploads/us-east-2:62eba507-efdc-48b7-85a9-2f09d417847c/2021/07/22/blogSN2.png"/>Figure 2: Logarithmic error in the ground energy level in the presence of shot noise for shot numbers 10, 50, 100, and 1000. One thing we learned from implementing shot noise in blueqat is that blueqat relies mostly on the backend for noise and does not implement its own noise simulation at all. Noise simulation is an unavoidable process in the simulation of quantum computation. The number of calculations on the actual machine is either set as a free usage limit or on a pay-as-you-go basis. For this reason, we really want to have a function that can reproduce the calculation of the actual machine for free. Shot noise alone is not enough to reproduce the noise in the actual machine, so I would like to take this opportunity to ask for the implementation of a function to simulate noise with detailed settings such as interaction noise, flip, phase shift, etc. [1] Ibe, Y., Nakagawa, Y. O., Yamamoto, T., Mitarai, K., Gao, Q., &amp; Kobayashi, T. (2020).

Implementation of shotnoise on blueqat

Hikaru Wakaura