Asking for a proof of an exponential speedup is a lot to ask for! We would need to show that all possible classical algorithms are incapable of efficiently simulating a quantum computer. This is really hard. But if we have a concrete family of classical algorithms, then surely we should try to understand if they could possibly offer an efficient simulation!
This brings us to stabiliser rank and related simulation methods. Let’s define some terms. Given a state we define the stabiliser rank to be the smallest number of terms needed to write as a sum of stabiliser states. More formally,
for some -qubit stabilizer states and some complex coefficients .
These decompositions were first studied by Garcia- Markov-Cross and applications to simulation were first explored in more depth by Bravyi-Smith-Smolin. The Gottesman-Knill theorem allows us to efficiently simulate a Clifford circuit acting on a stabiliser state. This can be extended to simulation of an arbitrary state undergoing a Clifford circuit with runtime proportional to (assuming we know the decomposition). We also know that a universal quantum computation can be gadgetized into a Clifford circuit with a supply of single qubit magic states where scales polynomially with the number of gates in the original quantum computation.
So if quantum computers are more powerful than classical computers, then we expect to scale exponentially with . If instead scales polynomially with then BQP=BPP (you can classically efficiently simulate quantum computers). But also the classical simulation method allows for postselection, so it would actually entail postBQP=BPP, which entails NP=P.
Surely, this is a simple statement that the quantum community should be able to prove. Surely, stabiliser rank simulators are not going to show BQP=BPP. Surely, surely we can prove the following conjecture:
holds for all integer .
Clearly, one has that , but the problem is that and that we have almost no lower bounds on the stabiliser rank. Consider, a 1-qubit non-stabiliser state
But when we consider 2 copies of the state we have
where
This is a decomposition with 3 stabiliser terms and so
for all single qubit . Ok, what about for more copies? Amongst many results in our recent Bravyi-Browne-Calpin-Campbell-Gosset-Howard (herein BBCCGH) is a proof that
for
(see Sec V.A of arXiv_V1). The stabiliser rank grows linearly! Not exponentially, but linearly! OK, this scaling holds up until only five copies, but I find this remarkable. Incidentally, the proof has a nice connection to the fact that the Clifford group forms a 3-design. Beyond five copies, the numerics look exponential, but the numerics are heuristic, inconclusive and only go slightly further than five copies!
Do we have any lower bounds on stabiliser rank! Deep in Bravyi-Smith-Smolin (Appendix C) is tucked away a neat result showing that for the state (an especially important single-qubit state) we know the stabiliser rank must scale at least as fast as for t copies. We are hoping for an exponential lower bound and yet the best we find is a square root. Not even a linear lower bound! I find this shocking. And I assure you, it is not for lack of many smart people spending many, many hours trying to get a better lower bound. In Sec V.D. of BBCCGH, we (I say “we” but this part was all Sergey) show an exponential lower bound but with an additional (and quite strong) constraint on the allowed stabiliser states in the decomposition. The proof technique is really interesting and hints at routes forward to solving the full problem.
There are also a whole bunch of cool open problems revolving around the approximate stabiliser rank (introduced by Bravyi-Gosset) and which BBCCGH show is connected to a quantity called “the extent” (see Def 3 of BBCCGH). I want to say more about these open problems but this post has already gone on too long. More to follow in part 2!
]]>Summary: The number of elementary gates in a quantum computation determines the runtime of the quantum computer. It is clearly advantageous to have faster computations that use fewer gates and “circuit compilation” is the art of optimizing and automating this process. For near-term quantum computers (without error correction) effective compilation is especially important because these devices will be noisy and this imposes a practical limit on the number of gates before an error becomes inevitable. Therefore, compilation protocols and software are crucial to whether we will be able to demonstrate a quantum advantage before full-blown error-corrected devices are available. This PhD project will develop compilation methods exploring random compilers and simulation of fermionic and bosonic interacting systems. This is a joint project between the University of Sheffield and Oxford University.
The studentship is fully funded for UK students with funding provided by NQIT, the UK national hub for quantum computing and Networked Quantum Information Technologies. As such, the student will have the opportunity to collaborate and contribute towards the UK’s largest quantum computing effort.
The project is a joint collaboration between the groups of Simon Benjamin in Oxford, and Earl Campbell in Sheffield. The project and student could be based at either university, according to the preference of the successful candidate. The studentship will be held at the chosen university and the student will be registered for a doctoral degree of the chosen university.
We are looking for an enthusiastic student with a physics, mathematics or computer science degree. The award should be a minimum of a UK upper second class honours degree (2:1). The student should also have some of the following desirable skills: a good undergraduate-level understanding of quantum mechanics, a strong mathematical background and/or experience programming and running numerical simulations (e.g. in C).
Informal inquiries can be made to: e.campbell@sheffield.ac.uk or simon.benjamin@materials.ox.ac.uk)
Formal applications can be made either to Sheffield via https://www.sheffield.ac.uk/postgradapplication/
or to Oxford University http://www.ox.ac.uk/admissions/postgraduate_courses/apply/index.html.
Here is a message from Dan about QEC ’19. It is coming to the UK!!! Very excited and pleased to be helping out with chairing the programme.
***
Dear colleagues,
I’m pleased to announce that the 5th International Conference on Quantum Error Correction (QEC’19) is now open for the submission of contributed talks and posters.
QEC ’19 will be taking place at Senate House, London between 29th July and 2nd August 2019. It aims to bring together experts from academia and industry to discuss theoretical, experimental and technological research towards robust quantum computation. Topics will include control, error correction and fault tolerance, and their interface with physics, computer science and technology research.
Please see our web-page for further information about the conference, including the list of invited speakers, and information on how to submit your one-page abstract: http://qec19.iopconfs.org/
The deadline for submissions is April 1st.
The scope of QEC is broad. Here is a non-exhaustive list of topics covered by the conference: Theoretical and experimental research in: Quantum error correcting codes; error correction and decoders; architecture and design of fault tolerant quantum computers; error mitigation in NISQ computations, error mitigation in quantum annealing; reliable quantum control, dynamical decoupling; Error quantification, tomography and benchmarking; Quantum repeaters; Active or self-correcting quantum memory. Applications of quantum error correction in Physics, CS and other disciplines; Classical error correction relevant to the QEC community; Compilers for fault tolerant gate sets; Error correction and mitigation in other Quantum Technologies (sensing / cryptography / etc.); QEC-inspired quantum computing theory.
If you have a query about whether your contribution might be in scope, please contact me directly.
Key dates
Abstract submission deadline: 1 April 2019
Registration opens: TBC (late February)
Early registration deadline: 10 June 2019
Registration deadline: 15 July 2019
A conference poster is available at the following link:
https://www.dropbox.com/s/tpz4v0q406kq1e0/QEC%202019%20London.pdf?dl=0
I would be grateful if you would print this poster and display it in a place where interested participants may see it.
I look forward to your submission to QEC’19 and to seeing you in London in the summer.
Regards,
Dan Browne – Conference Chair
]]>For many people, the gold standard is to use the fidelity. This is defined as
One of the main goals of this blog post is to explain why the fidelity is a terrible, terrible error metric. You should be using the “trace norm distance”. I will not appeal to mathematical beauty — though trace norm wins on these grounds also — but simply to what is experimentally and observationally meaningful. What do I mean by this? We never observe quantum states. We observe measurement results. Ultimately, we care that the observed statistics of experiments is a good approximation of the exact, target statistics. In particular, every measurement outcome is governed by a projector and if we have two quantum states and then the difference in outcome probabilities is
Error in outcome probabilities is the only physically meaningful kind of error. Anything else is an abstraction. The above error depends on the measurement we perform, so to get a single number we need to consider either the average or maximum over all possible projectors.
The second goal of this post is to convince you that randomness is really, super useful! This will link into the fidelity vs trace norm story. Several of my papers use randomness in a subtle way to outperform deterministic protocols (see here, here and most recently here). I’ve also written another blog posts on the subject (here). Nevertheless, I still find many people are wary of randomness. Consider a protocol for approximating that is probabilistic and prepares states with probability . What we will see is that the error (when correctly quantified) of probabilistic protocols can be considerably less than any of the pure states in this ensemble. Strange but true. Remember, I only care about measurement probabilities. And you should only care about measurement probabilities! Given a random protocol, the probability of a certain outcome will be
where
so our measure of error must depend on the averaged density matrix. Even though we might know after the event which pure state we prepared, the measurement statistics are entirely governed by the density matrix that averages over the ensemble of pure states. When I toss a coin, I know after the event with probability 100% whether it is heads or tails, but this does not prevent it from being an excellent 50/50 (pseudo) random number generator. Say I want to build a random number generator with a 25/75 split of different outcomes. I could toss a coin once: if I get heads then I output heads; otherwise I toss a second time and output the second toss result. This clearly does the job. We do not hear people object, “ah ha but after the first toss, it is now more likely to give tails so you algorithm is broken”. Similarly, quantum computers are random number generators and are allowed a “first coin toss” that determines what they do next.
Let’s thinking about the simplest possible example; a single-qubit. Imagine the ideal, or target state, is . We consider states of the form . The states and both have the same fidelity with respect to the target state. So too does the mixed state
.
Performing a measurement in the X-Z plane, we get the following measurement statistics.
We have three different lines for three states that all have the same fidelity, but the measurements errors behave very differently. Two things jump out.
Firstly, the fidelity is not a reliable measure of measurement error. For the pure states, in the worst case, the measurement error is 20 times higher than the error as quantified by fidelity. Actually, for almost all measurement angles the fidelity is considerably less than the measurement error for the pure states. Just about the only thing fidelity tells you is that there is one measurement (e.g. with ) such that the fidelity tells you the measurement error. But it is not generally the case that the quantum computer will perform precisely this measurement. This closes the case on fidelity.
Secondly, the plot highlights how the mixed state performs considerably better than the pure states with the same fidelity! Clearly, we should choose the preparation of the mixed state over either of the pure states. For each pure state, there are very small windows where the measurement error is less than for the mixed state. But when you perform a quantum computation you will not have prior access to a plot like this to help you decide what states to prepare. Rather, you need to design your computation to provide a promise that if your error is (correctly quantified) then all probabilities should be correct up-to and not exceeding . The mixed state gives the best promise of this kind.
Now we can see that the “right” metric for measuring errors should be something like:
Nothing could be more grounded in experiments. This is precisely the trace norm (also called the 1-norm) measure of error. Though it needs a bit of manipulation to gets into it’s most recognisable form. One might initially worry that the optimisation over projections is tricky to compute, but it is not! We rearrange as follows
let
where we have given the eigenvalue/vector decomposition of this operator. The eigenvalues will be real numbers because is Hermitian.
It takes a little bit of thinking to realise that the maximum is achieved by a projection onto the subspace with positive eigenvalues
Because, is traceless, the eigenvalues sum to zero and we can simplify this as follows
If we calculated this quantity and added it to the plots above, we would indeed see that it gives a strict upper bound on the measurement error for all measurements.
We have just been talking about measurement errors, but have actually derived the trace norm error. Let us connect this to the usual mathematical way of introducing the trace norm. For an operator , we denote or for the trace norm, which simply means take the absoluate sum of the eigenvalues of . Given two states, the trace norm distance is defined as
Hopefully, the reader can see that this is exactly what we have found above. In other words, we have the equivalence
Note that we need the density matrix to evaluate this error metric. If we have a simulation that works with pure states, and we perform statistic sampling over some probability distribution to find the average trace norm error
then this will massively over estimated the true error of the density matrix
I hope this convinces you all to use the trace norm in future error quantification.
I want to finish by tying this back into my recent preprint:
arXiv1811.08017
That preprint looks not at state preparation but the synthesis of a unitary. There I make use of the diamond norm distance for quantum channels. The reason for using this is essentially the same as above. If we have a channel that has error in diamond norm distance, it ensures that for any state preparation routine the final state will have no more than error in trace norm distance. And therefore, it correctly upper bounds the error in measurement statistics.
I gave a talk on random compiling that you can watch here:
https://collegerama.tudelft.nl/Mediasite/Play/ea10646f4d494cdaac3b17207d68cf601d?playfrom=0&player=0381c08c03db488d8bdbd4a5dfd3478b0a
The relevant paper is published here in Phys. Rev A. After giving the talk, I wrote a pair of blog posts (post 1 and post 2) to address the most common questions that arose during QIP coffee breaks.
with constants depending on the details. In my talk, I asserted that: (1) for qubit single gate sets, the Solovay-Kitaev algorithm achieved ; and (2) modern tools can efficiently solve the single-qubit problem using Clifford+T gates and achieve .
I further speculated that for multiqubit compiling problems, the optimal scaling (and hence value) could be much worse! But a few results have been pointed out to me. Firstly, the famous Dawson-Nielsen paper actually shows that is achieved by Solovay Kitaev for all compiling problems, and so is independent of the number of qubits. Secondly, this neat paper by Harrow-Recht-Chuang showed that an optimal compiler will always achieve scaling independent of the number of qubits.
However, the Harrow-Recht-Chuang result is non-constructive, so it doesn’t give us a compiler. It just says an optimal one exists. Also, it doesn’t say anything about how the classical runtime scales with the number of qubits. Therefore, if we restrict to compilers with polynomial runtime (polynomial in ), all we can say is that the optimal scaling sits somewhere in the interval . Finding where the optimal rests in this interval (and writing the software for a few qubit compiler) is clearly one of the most important open problems in the field.
The above discussion seems to entail that multiqubit compiling isn’t very different from single qubit compiling in terms of overhead scaling. However, we have a constant prefactor , which is constant with respect to but could increase with the number of qubits. Indeed, we know that there are classical circuits that need an exponential number of gates, which tells us that the prefactor for quantum compiler should also scale exponentially with qubit number.
]]>First, let’s recap. The setting is that we have some gate set where each gate in the set has a cost. If the gate set is universal then for any target unitary and any we can find some circuit built from gates in such that the distance between the unitaries is less than . 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 for some constants and depending on the details (see Part 2 for details).
The main result I presented was that we can find a probability distribution of circuits such that the channel
is close to the target unitary even though the individual circuits have cost upper bounded by . 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 and the experimentalist know that this unitary has been performed. But this particular instance has an error no more than , rather than . Is it that each circuit is upper bounded by noise, but that somehow the typical or average circuit has cost . No! Because the theorem holds even when every unitary has exactly 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 where is a coherent noise term with small . However, these are just subcircuits of a larger computation. Therefore, we really want to implement some large computation
.
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
with some probability
.
OK, now let’s see what happens with the coherent noise terms. For the subcircuit we have
so the whole computation we implement is
We can conjugate the noise terms through the circuits. For instance,
where
.
Since norms are unitarily invariant we still have
Repeating this conjugation process we can collect all the coherent noise terms together
Using that the noise terms are small, we can use
where
Using the triangle inequality one has
.
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 . 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 , we can’t say much more. But if we remember 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:
.
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 rather than the bound above that limits the tails of the distribution. But this gives us a rough intuition that 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 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
]]>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.
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