QIP random circuits mystery resolved! Part 1

Thank you QIP audience! On Tuesday, I gave a presentation on this paper
Phys. Rev. A 95, 042306 (2017)
https://arxiv.org/abs/1612.02689
I had some great questions, but in retrospect don’t think my answers were the best. Many questions focused on how to interpret results showing that random circuits improve on purely unitary circuits. I often get this question and so tried to pre-empt it in the middle of my talk, but clearly failed to convey my point. I am still getting this question every coffee break, so let me try again. Another interesting point is how the efficiency of an optimal compiler scales with the number of qubits (see Part 2). In what follows I have to credit Andrew Doherty, Robin Blume-Kohout, Scott Aaronson and Adam Bouland, for their insights and questions. Thanks!

First, let’s recap. The setting is that we have some gate set \mathcal{G}  where each gate in the set has a cost. If the gate set is universal then for any target unitary V  and any \epsilon > 0 we can find some circuit U = G_1 G_2 \ldots G_n  built from gates in \mathcal{G}  such that the distance between the unitaries is less than \epsilon . For the distance measure we take the diamond norm distance because it has nice composition properties. Typically, compilers come with a promise that the cost of circuit is upperbounded by some function f(\epsilon) = A \log ( 1 / \epsilon)^\gamma  for some constants A  and \gamma  depending on the details (see Part 2 for details).

The main result I presented was that we can find a probability distribution of circuits \{ U_k , p_k \} such that the channel

\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger

is O(\epsilon^2) close to the target unitary V  even though the individual circuits have cost upper bounded by f(\epsilon) . So using random circuits gets you free quadratic error suppression!

But what the heck is going on here!? Surely, each individual run of the compiler gives a particular circuit U_k  and the experimentalist know that this unitary has been performed. But this particular instance has an error no more than \epsilon , rather than O( \epsilon^2) . Is it that each circuit is upper bounded by \epsilon noise, but that somehow the typical or average circuit has cost O(\epsilon^2) . No! Because the theorem holds even when every unitary has exactly \epsilon error. However, typicality does resolve the mystery but only when we think about the quantum computation as a whole.

Each time we use a random compiler we get some circuit U_k = V e^{i \delta_k} where e^{i \delta_k} is a coherent noise term with small || \delta_k || \leq O(\epsilon) . However, these are just subcircuits of a larger computation. Therefore, we really want to implement some large computation

V^{(n)} \ldots V^{(2)} V^{(1)} .

For each subcircuit compiling is reasonable (e.g. it acts nontrivially on only a few qubits) but the whole computation acts on too many qubits to optimally compile or even compute the matrix representation. Then using random compiling we implement some sequence

U^{(n)}_{a_n} \ldots U^{(2)}_{a_2} U^{(1)}_{a_1}

with some probability

p_{a_n}^{(n)} \ldots p^{(2)}_{a_2} p^{(1)}_{a_1} .

OK, now let’s see what happens with the coherent noise terms. For the k^{th} subcircuit we have
U^{(k)}_{a_k} = V^{(k)} e^{i \delta_{a_k}^{(k)}}

so the whole computation we implement is

U^{(n)}_{a_n} \ldots U^{(2)}_{a_2} U^{(1)}_{a_1} = V^{(n)} e^{i \delta_{a_1}^{(n)}}\ldots V^{(2)} e^{i \delta_{a_2}^{(2)}} V^{(1)} e^{i \delta_{a_n}^{(n)}}

We can conjugate the noise terms through the circuits. For instance,

e^{i \delta_{a_2}^{(2)} } V^{(1)} =  V^{(1)} e^{i \Delta_{a_2}^{(2)}}

where

\Delta_{a_2}^{(2)}= V^{(1)} \delta_{a_2}^{(2)} (V^{(1)})^\dagger .

Since norms are unitarily invariant we still have

||\Delta_{a_2}^{(2)}|| = || \delta_{a_2}^{(2)} || \leq O(\epsilon)

Repeating this conjugation process we can collect all the coherent noise terms together

U^{(n)}_{a_n} \ldots U^{(2)}_{a_2} U^{(1)}_{a_1} = (V^{(n)} \ldots V^{(2)} V^{(1)} ) ( e^{\Delta_{a_n}^{(n)}} \ldots e^{\Delta_{a_2}^{(2)}} e^{\Delta_{a_1}^{(1)}})

Using that the noise terms are small, we can use

e^{\Delta_{a_n}^{(n)}} \ldots e^{\Delta_{a_2}^{(2)}} e^{\Delta_{a_1}^{(1)}} \sim e^{\Delta}

where

\Delta = \sum_k \Delta_{a_k}^{(k)}

Using the triangle inequality one has

|| \Delta ||  \leq \sum_k  || \Delta_{a_k}^{(k)}|| \leq n O(\epsilon)  .

But this noise term could be much much smaller than this bound implies. Indeed, one would only get close to equality when the noise terms coherently add up. In some sense, our circuits must conspire to align their coherent noise terms to all point in the same direction. Conversely, one might find that the coherent noise terms cancel out, and one could possibly even have that \Delta = 0 . This would be the ideal situation. But we are talking about large unitary. Too large to compute otherwise we would have simulated the whole quantum computation. For a fixed \Delta , we can’t say much more. But if we remember \Delta comes from a random ensemble, we can make probabilistic arguments about its size. A key point in the paper is that we choose the probabilities such that the expectation of each random term is zero:

\mathbb{E} (  \Delta_{a_k}^{(k)} )=   \sum_{a_k} p^{(k)}_{a_k} \Delta_{a_k}^{(k)} = 0  .

Furthermore, we are summing a series of such terms (sampled independently). A sum of independent random variables are going to convergence (via a central limit theorem) to some Gaussian distribution that is centred around the mean (which is zero). Of course, there will be some variance about the mean, but this will be \sqrt{n} \epsilon  rather than the n \epsilon  bound above that limits the tails of the distribution. But this gives us a rough intuition that \Delta will (with high probability) have quadratically smaller size. Indeed, this is how Hastings frames the problem in his related paper arxiv1612.01011. Based on this intuition one could imagine trying to upper bound \mathbb{E} (  || \Delta|| )  and make the above discussion rigorous. Indeed, this might be an interesting exercise to work through. However, both Hastings and I instead tackled the problem by showing bounds on the diamond distance of the channel, which implicitly entails that the coherent errors are composing (with high probability) in an incoherent way!

More in part 2

QCDA consortium

Last year the EU recommended our QCDA (Quantum Code Design & Architectures) network for funding via its QuantERA programme! The consortium has some really amazing scientists involved and we will be recruiting 5 more people to join (4 postdocs and 1 PhD student). If you want to learn more about the network, I’ve set up a website dedicated to QCDA

http://www.qcda.eu/

We are still waiting to hear from the national funding agencies when the project can start but it could be as soon as February 2018.

ThinkQ

ThinkQ hosted by IBM was a really outstanding meeting. You can watch almost all of the talks online here.

I gave a talk on classical simulation of quantum circuits. As usual, I’ll post my talk here

QIP talks online

The videos for QIP talks are online now. Actually, they have been online for a while but I’m only just posting about it. The talks are hosted on the Microsoft Youtube channel and here is a playlist
https://www.youtube.com/playlist?list=PLq2e2Zi6KW-vedSEsTvM81xrbB7WnfTz5

Below I’ve embedded the two QIP talks by myself and Mark Howard on our recent work together.

QIP talk by Earl Campbell based on work published in
Phys. Rev. Lett. 118 060501 (2017)
Phys. Rev. A 95 022316 (2017)

QIP talk by Mark Howard based on work published in
Phys. Rev. Lett. 118 090501 (2017)

Not long until QIP

Next week the annual edition of QIP is hosted by Microsoft in Seattle. QIP is always really exciting, full of great talks, lots of familiar and new faces. Though I’m especially looking forward to it this year. For me it is the first QIP hosted by a multinational technology company. I just googled and it seems IBM hosted QIP back in 2002, but that was way back in my undergrad day. Also, my first time to Seattle, home of Grunge music, one of my teenage obsessions.

The last few days I have been working on my slides for QIP where I will be a talking about work with Mark Howard on Unifying gate-synthesis and magic state distillation. We were doubly lucky this year as Mark will also be talking about our joint work on a resource theory of magic states.

For anyone who wants a short summary of these papers, I thought it would be nice to post our QIP extended abstracts here, which were written with more of a QIP audience (rather than APS journal audience) in mind. Enjoy.

QIP EXTENDED ABSTRACT: UNIFYING

QIP EXTENDED ABSTRACT: RESOURCE