Unitary 2-designs, variational quantum eigensolvers, and barren plateaus

Are randomised quantum circuits inducing unitary 2-designs a reasonable thing to hope for in the near future? It seems so, following a paper presented by Saeed Mehraban at QIP this year. With Harrow, Mehraban showed that short-depth random circuits involving nearest-neighbour unitary gates (where proximity is defined by a cubic lattice structure) produce approximate unitary $t$-designs. This is precisely the experimental model of Google’s Quantum AI group, who are working with a 49-qubit 2-dimensional lattice of qubits.

The main application of these randomised circuits is to prove ‘quantum supremacy’. But can they do anything useful? In the attached note, I discuss an application to building a variational quantum eigensolver, which runs into the problem of ‘barren plateaus’ recently highlighted by McClean et al. (See within for references.)


3 thoughts on “Unitary 2-designs, variational quantum eigensolvers, and barren plateaus

    1. Glad you found it helpful!

      Not sure if it’s relevant to what you’re thinking about, but I’ve been following up on this with Kenji Capanelli, an MSc student here at Bristol. In particular, we were wondering whether, given some parametrised circuit, one could actually define the path in the unitary group resulting from VQE without talking about the parametrisation at all.

      To formulate gradient descent on the unitary group requires a bit more structure than was necessary to demonstrate the barren plateaux problem. There we only cared about the scaling with n of directional derivatives in a single tangent space. Here, though, we need a Riemannian metric in order to define the unit ball in the tangent space, and the exponential curve along a particular tangent vector.

      We get a surprising clash between gradient descent in parameter space (Euclidean metric on R^n) and gradient descent in the unitary group, since there’s no way to ‘push forward’ the Euclidean metric along the parametrisation map in general. The path in the unitary group induced by gradient descent in parameter space could be some wild meander which does not make sense according to any metric on the unitary group!

      There’s lots of interesting things still to prove. Gradient descent in the unitary group is the natural notion and should be ‘better’ than gradient descent using any parametrisation. Formalising and proving ‘better’ could be interesting – for starters, which of the many metrics on the unitary group is ‘best’? One might hope to apply this to upper bound VQE runtimes, or optimising circuit parametrisations. All a nice excuse for some basic differential geometry πŸ™‚

  1. Thanks a lot for the comments – sounds very interesting! πŸ™‚
    I’m still reading up and learning this area but might send over some more detailed ideas in a few days.

    One specific question I had:

    Consider the map representing the whole variational process, taking some parameter set to an energy expectation value. (This map can be decomposed into mapping parameters to the unitaries, mapping the unitaries to a state, then mapping a state to it’s expected energy). We can differentiate this total map explicitly with repeated use of:

    d/dx ( e^{ixH} ) = iHe^{ixH}

    to get a gradient of the whole variational process.

    Although I’m very interested in lie groups/algebras/diff geom and their applications to QI, I’m wondering what the benefit could be of isolating the ‘unitary part’ of the process and considering the differentiability of (for example) the map: U(n) –> Energy. Two reasons I can think of are:
    1) to be able to use results about the Haar measure in considering the induced distribution from the parameters on the unitaries
    2) to better understand the geometry/trajectory of a variational process

    In short my question is: what do you mean by ‘gradient descent on the unitaries’, and ‘without talking about the parametrisation at all’? How would this be useful?

    Thanks πŸ™‚

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