Changed expect_g and fidelity_g to return the "correctly" scaled gradient w.r.t to the register

[jlbosse]

Up until now expect_g and fidelity_g were off by a factor of one half, in the sense that ∂_reg expect(ham, reg) = 2 * expect_g(ham, reg) instead ∂_reg expect(ham, reg) = expect_g(ham, reg) and analogous for the other methods of expect_g and fidelity_g. This was counter acted by multiplying a factor of 2 to the final paramsδ at the end of expect_g(ham, reg => circuit) and fidelity_g(reg1 => circuit1, reg2 => circuit2).

This is changed here to backpropagate the "correctly" scaled gradient w.r.t registers.

I am not sure if this would be considered a breaking change, because it also means that the output of expect'(ham, reg => circuit)[1] is changed by a factor of 2 now.

Changes:

• Changed expect_g and fidelity_g
• Updated the rrules in chainrules_patch.jl
• Updated the corresponding tests

[codecov[bot]]

Codecov Report

Base: 88.28% // Head: 86.93% // Decreases project coverage by -1.35% :warning:

Coverage data is based on head (30c388a) compared to base (9d7ab2e). Patch has no changes to coverable lines.

Additional details and impacted files

```diff @@ Coverage Diff @@ ## master #446 +/- ## ========================================== - Coverage 88.28% 86.93% -1.36% ========================================== Files 76 16 -60 Lines 4765 1408 -3357 ========================================== - Hits 4207 1224 -2983 + Misses 558 184 -374 ``` | [Impacted Files](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS) | Coverage Δ | | |---|---|---| | [lib/YaoBlocks/src/autodiff/chainrules\_patch.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvYXV0b2RpZmYvY2hhaW5ydWxlc19wYXRjaC5qbA==) | | | | [lib/YaoBlocks/src/autodiff/specializes.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvYXV0b2RpZmYvc3BlY2lhbGl6ZXMuamw=) | | | | [lib/YaoBlocks/src/measure\_ops.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvbWVhc3VyZV9vcHMuamw=) | | | | [src/EasyBuild/hamiltonians.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-c3JjL0Vhc3lCdWlsZC9oYW1pbHRvbmlhbnMuamw=) | | | | [lib/YaoBlocks/src/utils.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvdXRpbHMuamw=) | | | | [lib/YaoBlocks/src/abstract\_block.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvYWJzdHJhY3RfYmxvY2suamw=) | | | | [lib/YaoBlocks/src/treeutils/treeutils.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvdHJlZXV0aWxzL3RyZWV1dGlscy5qbA==) | | | | [src/EasyBuild/qft\_circuit.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-c3JjL0Vhc3lCdWlsZC9xZnRfY2lyY3VpdC5qbA==) | | | | [lib/YaoBlocks/src/treeutils/dump.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvdHJlZXV0aWxzL2R1bXAuamw=) | | | | [lib/YaoBlocks/src/treeutils/load.jl](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS#diff-bGliL1lhb0Jsb2Nrcy9zcmMvdHJlZXV0aWxzL2xvYWQuamw=) | | | | ... and [50 more](https://codecov.io/gh/QuantumBFS/Yao.jl/pull/446/diff?src=pr&el=tree-more&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS) | | Help us with your feedback. Take ten seconds to tell us [how you rate us](https://about.codecov.io/nps?utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS). Have a feature suggestion? [Share it here.](https://app.codecov.io/gh/feedback/?utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=QuantumBFS)

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[Roger-luo]

need tests pass before a review

[jlbosse]

The tests fun on my local machines on julia 1.8.3 and 1.8.2 and are in parts of the tests that shouldn't be affected by my changes at all. So I also think that the failures on nightly are unrelated.