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I'm studying output expectation values of random quantum circuits. These random circuits are built using random gates from a universal set. I have noticed that the distribution of the output expectation values (of random Pauli observables) of different quanum circuits have mean close to 0 and variance that goes to 0 as the number of qubits increase (I don't know if there is a rigorous explanation for this).

My question is: is there a class of quantum circuits that does not show this behaviour? I'm interested in quantum circuits that have some kind of "parameter" that I can change since I need different instances.

For example, I think that a quantum circuit representing the last step of a VQE (Variatonal Quantum Eigensolver) process should have some expectation value far from 0 (i.e. the ground state energy) since it represents the ground state of an Hamiltonian (but it would be too expensive to run an entire VQE algorithm for each instance). Instead, if I use random angles (without particular constraints) for the same quantum circuit ansatz, I have the same behaviour described at the beginning of the post.

Thank you.

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    $\begingroup$ What you are describing is that your random circuits are behaving too "Haar-randomly" and yes, there is a rigorous explanation for this using approximate unitary 2-designs. To avoid this, either you do not allow your circuits to be universal (or 2-design-ish), or you make them very shallow. $\endgroup$ Nov 7, 2023 at 12:24
  • $\begingroup$ @MarkusHeinrich Thank you for your answer. I would like to have quantum circuits that are difficult to simulate for classical simulation techniques (also tensor networks), so they shouldn't be shallow. Do you have in mind any categories of quantum circuits that have these properties? $\endgroup$
    – stopper
    Nov 7, 2023 at 13:45
  • $\begingroup$ I see. Just that I understand you right: Do you compute mean and variance over different random circuits or over random Paulis? $\endgroup$ Nov 7, 2023 at 14:03
  • $\begingroup$ @MarkusHeinrich over different random quantum circuits. $\endgroup$
    – stopper
    Nov 7, 2023 at 14:15

1 Answer 1

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Use circuits 15 and 6 from this paper. They are random (parametrized) and scalable, i.e. you can make them as deep as you want, and, at the same time, they are anti-'Haar random'.

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  • $\begingroup$ Thank you. I tried them right now but they have the same behaviour of random quantum circuits, at least for the expectation values of single qubit Z observables. I mean, the distribution of the output expectation values is peaked at 0 for large number of qubits (i.e. small variance). $\endgroup$
    – stopper
    Nov 7, 2023 at 17:19
  • $\begingroup$ What kind of distribution you want to have? Please specify. "I mean, the distribution of the output expectation values is peaked at 0 for large number of qubits (i.e. small variance)." - so you have probability concentration on the binary strings where the number of zeros is substantially larger than $n/2$? $\endgroup$
    – trurl
    Nov 7, 2023 at 17:54
  • $\begingroup$ I'm not interested in a particular kind of distribution, but when I deal with random quantum circuits I have this problem: if I execute many random circuits and I calculate expectation values of an observable (e.g. Pauli) of these circuits, the distribution of these output expectation values have mean close to 0 and variance that decreases as the number of qubits increases (they become more peaked in 0 as the number of qubits increases). I want quantum circuits that don't have this behaviour, since at this point many expectation values are really close to 0 and it is not interesting to study. $\endgroup$
    – stopper
    Nov 7, 2023 at 18:25
  • $\begingroup$ For the second question "you have probability concentration on the binary strings where the number of zeros is substantially larger than $n/2$?" No, if this was the case the expectation values would have been close to 1 (since the eigenvalue of $Z$ for $|0\rangle$ is 1). $\endgroup$
    – stopper
    Nov 7, 2023 at 18:27
  • $\begingroup$ Then you need circuits which yield heavy-tailed distributions - yet they (i.e., distributions) are also 'peaked' but their tales are heavier then usual. Otherwise, put at the end of your circuit a layer of a single-qubit Hadamard gates and enjoy the spread. $\endgroup$
    – trurl
    Nov 7, 2023 at 18:44

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