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The logic for stddev_upper_bound is the same as at https://github.com/Qiskit/qiskit-ibm-runtime/blob/ce2e94028bfd60ce99c2fa1d5a9d56601e481e34/qiskit_ibm_runtime/utils/sampler_result_decoder.py#L32-L44

Maybe @t-imamichi or @nonhermitian can comment on the correctness. The stddev_upper_bound attribute is poorly documented, but my understanding is that it represents an upper bound on the standard deviation of the expected value of the operator ZZ...Z.

Ok, but if it were a bound on the stddev of the ZZ... operator then assume N=2 shots, measuring once +1 and once -1. Then you have a stddev of 1 but the upper bound would be 1/N = 1/2? That's why I'm a little confused about this argument...

Consider a quantum state |ψ⟩ for which the operator ZZ...Z is measured N times, yielding a collection of N numbers [+1, +1, -1, ...]. The confusion arises because there are two distinct quantities that may both be referred to as the "standard deviation":

1. The standard deviation of the collection of N numbers [+1, +1, -1, ...].
2. Let A be the average of the N numbers [+1, +1, -1, ...]. Then A can be considered to be drawn from a probability distribution P representing the result of an experiment in which |ψ⟩ is prepared and measured N times and the results are averaged (repeating this experiment would yield different values for A). The "standard deviation" could refer to the standard deviation of P.

In your comment above you refer to (1), but stddev_upper_bound refers to (2). Also, note that it's 1/sqrt(N), not 1/N.

Because the output can be a quasi-distribution with negative values one cannot use standard variance formulas. One can however get an upper-bound on this as described here: https://arxiv.org/abs/2006.14044,
valid only for operators |O| <= 1

Because the output can be a quasi-distribution with negative values one cannot use standard variance formulas. One can however get an upper-bound on this as described here: https://arxiv.org/abs/2006.14044, valid only for operators |O| <= 1

In this case it's a probability distribution though, since it's calculated as
https://github.com/Qiskit/qiskit-terra/blob/b271adb3087205f05cbc11288e582dfd20a48fa7/qiskit/primitives/backend_sampler.py#L169

In this case it's a probability distribution though, since it's calculated as

https://github.com/Qiskit/qiskit-terra/blob/b271adb3087205f05cbc11288e582dfd20a48fa7/qiskit/primitives/backend_sampler.py#L169

Yeah for the BackendSampler it's returning a normal probability distribution because there is no error mitigation being done as part of the generic implementation for any backend it does. It just computes the probability via division, we returned a QuasiDistribution because that's what the abstract sampler api dictates https://github.com/Qiskit/qiskit-terra/blob/c2215ae87b516e57357075d69753489f056cbabc/qiskit/primitives/base/sampler_result.py#L44

Pull Request Test Coverage Report for Build 3951860569

• 4 of 4 (100.0%) changed or added relevant lines in 1 file are covered.
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Totals
Change from base Build 3951859300:	0.08%
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