Monday, June 30, 2014

Sampling arbitrary photon-added or photon-subtracted squeezed states is in the same complexity class as boson sampling



Jonathan P. Olson, Kaushik P. Seshadreesan, Keith R. Motes, Peter P. Rohde, Jonathan P. Dowling

Boson sampling is a simple model for non-universal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single photon states is fed through a Haar-random linear optics network and sampled at the output using coincidence photodetection. This problem is strongly believed to be classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. The equivalence is exact and independent of the squeezing parameter, and hence provides an entire class of new quantum states of light in the same complexity class as boson sampling.

See: arXiv:1406.7821

Thursday, June 26, 2014

An introduction to boson-sampling

Bryan T. Gard, Keith R. Motes, Jonathan P. Olson, Peter P. Rohde, Jonathan P. Dowling

Boson-sampling is a simplified model for quantum computing that may hold the key to implementing the first ever post-classical quantum computer. Boson-sampling is a non-universal quantum computer that is significantly more straightforward to build than any universal quantum computer proposed so far. We begin this chapter by motivating boson-sampling and discussing the history of linear optics quantum computing. We then summarize the boson-sampling formalism, discuss what a sampling problem is, explain why boson-sampling is easier than linear optics quantum computing, and discuss the Extended Church-Turing thesis. Next, sampling with other classes of quantum optical states is analyzed. Finally, we discuss the feasibility of building a boson-sampling device using existing technology.

See: arXiv:1406.6767

Friday, June 20, 2014

Scalable boson-sampling with time-bin encoding using a loop-based architecture

Keith R. Motes, Alexei Gilchrist, Jonathan P. Dowling, Peter P. Rohde

We present an architecture for arbitrarily scalable boson-sampling using two nested fiber loops. The architecture has fixed experimental complexity, irrespective of the size of the desired interferometer, whose scale is limited only by fiber and switch loss rates. The architecture employs time-bin encoding, whereby the incident photons form a pulse train, which enters the loops. Dynamically controlled loop coupling ratios allow the construction of the arbitrary linear optics interferometers required for boson-sampling. The architecture employs only a single point of interference and may thus be easier to stabilize than other approaches. The scheme has polynomial complexity and could be realized using demonstrated present-day technologies.

See: arXiv:1403.4007

A Quantum Phase Representation of Heisenberg Limits and a Minimally Resourced Quantum Phase Estimator


Scott Roger Shepard, Frederick Ira Moxley III, Jonathan P. Dowling

Within the quantum phase representation we derive Heisenberg limits, in closed form, for N00N states and two other classes of states that can outperform these in terms of local performance metrics relevant for multiply-peaked distributions. One of these can also enhance the super-resolution factor beyond that of a N00N state of the same power, at the expense of diminished fringe visibility. An accurate phase estimation algorithm, which can be applied to the minimally resourced apparatus of a standard interferometer, is shown to be resilient to the presence of additive white-Gaussian noise.



See: arXiv:1403.2313

On The Future of Quantum Computing as Seen from the Recent Past


During the period 2011–2013 I was honored to participate as a co-investigator on one of the four teams funded under the Intelligence Advanced Research Projects Activity (IARPA) Quantum Computer Science Program (QCS). To quote from the IARPA web page, “The IARPA Quantum Computer Science (QCS) Program explored questions relating to the computational resources required to run quantum algorithms on realistic quantum computers.”

First let me state that this blog post represents my own viewpoint and opinion of the program and not that of IARPA or other participating teams or government agencies involved. However, as most of my readers know, given my involvement in the US Department of Defense (DoD) and Intelligence Agency programs in quantum computing from their inception in 1994, both as an advisor to the government and as a government-funded researcher, I perhaps have a unique perspective that, hopefully, can help us see the big picture of what emerged from the QCS program. (For readers who don’t know my background in this area, please pick up a copy of my new book, Schrödinger’s KillerApp: Race to Build the World’s First Quantum Computer, and then very carefully set it down again.)

Quantum computing is not merely the sum of a qubit hardware technology and quantum control and quantum error correction and an algorithm, but rather is a system where the combination of those elements may work better or worse than the sum of the parts.  This, of course, leads to the research questions related to where those nonlinearities are, how to avoid or exploit them as appropriate, and how to build tools to allow easier control of those nonlinearities. Our particular team’s work on decoherence-free subspaces, in this context, was an early example of what we hoped to achieve in that it ignores the line between quantum computing and quantum error correction and focuses on how to build a protocol that performs both functions in an integrated way better than you'd get with individual pieces.

Particularly I would first like to spell out that our team was a unified and integrated research program and not just a loose federation of independent research efforts. Frankly, going into this, I thought this unification would be difficult, if not impossible, to attain. My fear was that our team, consisting of tens of researchers from backgrounds in classical computer science, quantum physics, engineering, and so forth would have a great deal of trouble even communicating with each other — to get beyond the specialty specific jargon issue. This was perhaps a problem in the first few months but we all learned from each other very rapidly and by the end of the first few months of Phase I, everybody was on the same plane and had a much better view of the integrated whole than, say, when we wrote the original proposal. Sometimes throwing a bunch of people from disparate backgrounds into a room and leaving them to figure out how to communicate actually does work.

As this intra-team communication progressed, I personally came to better understand in just what ways quantum computer is more than the sum of its individual parts. That is a quantum computer is not just a particular qubit hardware platform that is cobbled together with quantum control, quantum error correction, quantum compilers and programming language. For example, as a theoretical physicist that has worked close to the qubit hardware for many years, it was difficult for me wrap my brain around what a quantum programming language would even look like. Hence it was quite comforting to me, when I met our team member Peter Selinger in person for the first time at the kick-off meeting, that he confessed to me that he, in spite of taking the lead, both internationally and on our team, in developing exactly such a programming language (Quipper), that he had trouble wrapping his brain around the some of the things we found in the government furnished information such as the description of errors in terms of Krauss operators. Hence it was that, while our team had experts in all of the disparate sub-areas needed to carry out the requisite resource estimates, at least initially I am sure none of us really saw the big picture. That changed very quickly as the project began to pick up steam and our team was really able to integrate these disparate sets of expertise so that what we were able to produce was type of interactive and nonlinear system analysis where the various pieces of the quantum computer were being tailored in real time to work well with all the other pieces in a way that at the outset I had not expected to happen so quickly.

An outsider once asked me what the QCS program was all about and, somewhat flippantly, I replied that it was the Manhattan Project for quantum computing. (The only difference is that, for the Manhattan Project, the end goal was to ensure that something blew up but for the QCS program the end goal was to make sure that nothing ‘blew up’.) In hindsight I do no think this remark is flippant at all. Surely we were not, unlike the Manhattan Project, trying to build an actually working device, but like the Manhattan Project we had experts from disparate fields all working on different aspects of the development of the resource estimation. Let me give a particular example. Going into this program, no one our team really understood the interplay between the resource estimates, the type of quantum error correction (QEC) code employed, and the type of error mitigation techniques such as quantum control (QC) and decoherence free subspaces (DFS) that needed to be used. We all understood that there was going to be a tremendous amount of overhead coming from the QEC codes, particularly if the errors where near threshold where large numbers of concatenation would be required, but we did not really have a quantitative feel for how this all played out. As the baseline resource estimates of our team and the other three teams began to give these initially somewhat long time scales for running various algorithms, we were then able to quantitatively go back and investigate the quantum error mitigation methods with a much better understanding. As it was unclear which mitigation would work best with which government-provided physical machine description (PMD) we tried a number of them in parallel including active quantum control with feedback, dynamical decoupling, and DFS. This is similar to, for example, in the Manhattan Project where they pursued both uranium and plutonium designs in parallel, as well as spherical implosion and gun designs for inducing detonation. One of the things that we quickly realized is that for most of the PMDs the noise and decoherence was not well enough understood nor characterized in the experimental literature for the theorists to actually optimize over the differing mitigation protocols.

Another interesting result to come out of our teams ‘divide and conquer’ strategy was that, going into the program, I had the feeling the each of the QEC encodings would perform reasonably well. I would wager this was the assumption of many in the field. What clearly came out of this program is that some QEC encodings, for the particular PMDs investigate here, really do much better and across all PMDs. Although resistant to this idea at first, we finally did convince ourselves that what some vocal members of other teams had been saying all along was likely correct; surface codes were clearly the best shot at moving towards a scalable quantum computer with current levels of errors in the gates and the qubit storage. However we were able to quantify just in what way that was correct using the tools developed in our research program so we did not have to just take another team’s word for it. 

It is interesting to reflect on the role of error correction codes in the development of classical computers. In the 1950s VonNeumann (and others) developed rudimentary classical error correction codes to address the problem that the digital computers of the time, such as the ENIAC and the EDVAC, were blowing out vacuum tubes about as fast as they could replace them. These codes are not used today because of the technological growth curve that has taken us from vacuum tubes to the integrated circuit where fault tolerance is built into the hardware. To foresee that classical technology curve in the 1950s, VonNeumann would have to have predicted the invention of the integrated circuit (1960s), the Mead-Conway VLSI design rules (1980s), and the consequent path of Moore’s Law that has given us the fault-tolerant computers of today. In many ways the QCS program is much like an attempt to carry out resource estimates for various classical computer algorithms while assuming that the hardware is some variant of the ENIAC or other vacuum tube technology. Indeed, a single ion-trap quantum computer with millions or billions of qubits, along with all the requisite trapping and gate implementation hardware would easily fill a large warehouse and weight several tons, just as the ENIAC did in its day.

Very likely, if resource estimates for now common classical algorithms were carried out in the 1950s, the researchers would have also run into long time scales for their baseline resource estimates (for, say, the algorithm for playing World of Warcraft on an ENIAC) — particularly if large amounts of classical error correction needed to be deployed. Yet we now know that the then unforeseen breakthroughs leading up to Moore’s Law allowed us to reduce the size and weight of such a computer so that it fits in my pocket (iPhone) and the run time for most algorithms of interest are measured in milliseconds. The wrong message to take away from a baseline resource estimate made in the 1950s for classical computers was that classical computing was doomed. In the same way it is the wrong message to take away from the QCS program that the large baseline resource estimates that PLATO and all the teams found should be a sign that quantum computing is doomed. These baseline resource estimates (BRE) form an upper bound against which all unforeseen (but highly probable to be found) future technological advances in quantum computer hardware or architecture developed will be measured. All new technologies have initially a slow growth but inevitably new discoveries and innovations come to bear and the growth then settles into an exponential pace of development. There will be someday a quantum Moore’s Law as the these new discoveries are deployed and in tens of years the warehouse-sized quantum computer will fit in my new grandnephew’s pocket.

Our team’s BRE work came out of the need to understand the problem space and develop a baseline.  Most non-algorithm work that has considered quantum algorithms at all has looked at Shor's or possibly Grover's.  These algorithms do have the advantage that they are relatively simple and their practical value would be immediate.  However, if we presume that quantum computers are going to be generally useful, rather than dedicated problem solvers, then we also assume the space of algorithms will continue to grow and that we'll have other, more complex algorithms that are useful.  Focusing on the simplest algorithms has, in computer science, traditionally led to bad decisions about what is possible and what resources are necessary. Thus, the QCS program has examined a range of algorithms — from some that would be practical to run if a quantum computer of modest resources could be built — up to some that will likely never be practical to run. 

Thus the QCS program is similar to the classical computer algorithm analysis in HACKMEM performed in 1972 in “Proposed Computer Programs, In Order Of Increasing Running Time” (problem 77–96).  They consider several games up to checkers and what it would take to “solve” them.  Then they also consider hex, chess, and go; analyzing their complexity.  In particular their results would have seemed to indicate that chess and go would be intractable on a classical computer. In contrast to what some might take away from the QCS baseline resource estimates, the HACKMEN group did not, therefore conclude that classical computing is impossible or worthless.  Rather this sort of analysis is generally taken as being useful for having a broad view of what is and is not feasible as well as presenting the basis for understanding when heuristic short cuts will be necessary.  Thus, chess is an example of a game where classical computational brute force techniques are insufficient, so chess playing computers use a combination of techniques — from stored game openings and endings to search techniques that increase the probability of finding a good move in a reasonable amount of time at the possible expense of finding the best move. Contrary to what one might have taken away from the HACKMEM results in 1972, since 1997 the best chess players in the world are computers ever since IBM’s Deep Blue beat world chess champion Garry Kasparov.

The QCS BRE’s are also necessary as a baseline. For networking audiences, I typically mention that we have a baseline.  For networking programs, we’re either asked how much we can improve over TCP (for pure networking) or how much worse than TCP are we (for security-focused programs).  For quantum computing, we haven't had a similar baseline to compare against.  Even if it's imperfect, this is something that the community very much needs to make it easier to compare claims and to make it harder for people to report results that work only on their carefully chosen problem.  (It may be going to far to point out that various people have started referring to what D-Wave presents results against as "the D-Wave problem" since it was carefully chosen to match the capabilities of their machine.)

So what did we learn in by the end of the QCS program? Along the lines of the Manhattan Project analogy, we have learned that it is indeed possible to put together a team of smart people with disparate backgrounds ranging from physics to computer science and hammer out a common language and mode of thinking that allowed us to successfully make reasonable BREs for the wildly different PMDs and algorithms. Going into this at the kick off meeting I was not sure even that would be possible. And yet here we are at the end of Phase II and all four teams, taking often different approaches, found similar patterns for all of the algorithms investigated.

We learned in a quantitative way just how the bad the QEC overhead is and in just what way that overhead affects the various run times for different choices off the menu. Before QCS there was only a qualitative feeling for how this would play out. In particular QCS gave us a robust way to compare different QEC schemes to each other. Before QCS different researchers chose QEC schemes based on what was either popular at the time or what they were most familiar with. Particularly I was not convinced that one QEC, surface codes, would be dramatically better than the others — in spite of dramatic claims made by members of other teams. Where were the numbers to back up these claims? Well the QCS program produced them and our own numbers produced by our team told us that well indeed that surface error correcting codes were probably the way to go.

Another critical point that came out of the QCS program was the interplay between quantum error correction (QEC), quantum control (QC), and other error reduction methods such as decoherence free subspaces (DFS) and quantum dynamical decoupling (QDD). The physical machine descriptions (PMD) and government furnished information (GFI) was presented to us in such a way that, unless something was done, the native error rates in the hardware were all just right at the threshold for where the QEC would work. My understanding is that the government folks expected all the teams to use QC, DFS, and QDD to lower these given thresholds to a point where large depth QEC codes could be avoided. It did not become clear until the meeting in Princeton in July of 2012 that nearly all the noise models provided to the performing teams by the government teams were of a sort where quantum control techniques would not work.

The most critical point to come out at that meeting was that, in fact, in most of the quantum computing experiments these noise spectra are either unknown or unmeasured, which I suspect lead to the simplest assumption, which was to make them all Gaussian, precisely the type of noise that the most powerful quantum control methods are useless against. At one review meeting it was deemed important get the experimenters and the quantum controllers all into the same room for a two day meeting to hash out just what the theorist needed from the experiments to optimize the control techniques. Such a meeting, I think, would be still critical for the advancement of the field of quantum computing. It was, in my opinion, this being stuck right at the error correction threshold that led to the huge code depths and the long time scales of the resource estimates

It appears to me that rather than having a generic quantum control or other decoherence-mitigation methodology that would be applied to all hardware platforms; it will be critical in the short term to develop quantum control techniques that are specifically tailored to the noise spectrum of each of the physical devices. This suggests a program where the quantum control and QEC community sit down with the experimenters and hash out just what experiments need to be done and what type of data is needed in what form. Then next from these data specific noise spectra models would be then developed and finally the QEC and QC theorist would produce tailored quantum control techniques to match the noise spectra. Close collaboration with the experimentalists is also needed so that the QC theorists understand, particularly for active quantum control, just what types of weak and strong measurements are possible in each experimental implementation. For example, in a recent experiment with ion traps that demonstrated Ulrich dynamical decoupling (UDD) as a decoherence mitigation protocol, the experimenters deliberately added noise to their system, noise with a spectrum that UDD was designed to correct, and then corrected it. This is clearly not the way to go. The QC methods should be tailored to the native noise spectrum and not the reverse.

Another big success of the QCS program was the ability to exploit the Gottesman-Knill theorem in order to characterize errors in large-depth codes that cannot be handled by a straight forward Monte Carlo method due to the exponentially large Hilbert space. In the proposal writing stage in 2010 we had some vague ideas that something like this would be possible in that the gates in the Clifford group, while efficiently simulatable classically, might still span enough of the computational circuit space so that we could compute errors in large-depth circuits without having to build a quantum computer to do so. The concrete result was the resource-estimating tool (QASM-P) that we developed and I was really surprised how well this worked. Such a tool will be a mainstay for quantum circuit characterization for years to come, at least until we have enough quantum computers around where we can start using few-qubit machines to characterize few-more qubit machines in a bootstrapping approach. QASM-P was in fact just one of several stand-alone tools developed by our team on the fly to address particular computations as needed. What I find amazing is that even though these tools were developed by different team members; it was possible to integrate their usage across all the resource estimates and produce coherent results

This issue of large scale circuit performance characterization, to my mind, is the real roadblock for building large quantum computers. It is related to the issues that currently plague, for example, ion trap computers with just about 10 qubits. The known protocols for characterizing the gates and states are quantum process and quantum state tomography; protocols that require an exponentially large number of data points to be taken in the number of qubits. Blatt is fond of saying the he could easily build a 16 qubit ion trap register but the classical resources for characterizing the device’s performance are lacking. Here the Clifford algebra approach does not help. The example I like to use is that when Intel tries to characterize the performance of its latest chip, it does not do so using the ENIAC. Hence there will come a time when we will have to use few-qubit machines to characterize the performance of few-more qubit machines and in this way bootstrap our way up to many qubit machines. However this quantum-to-quantum bootstrapping characterization protocol does not yet exist, even in theory. Until we get around this problem we’ll be limited to machines of only 10–20 qubits by pursing the bottom up approach of adding a new qubit to the device every year or so.

So in summary, when I first read the Broad Agency Announcement (BAA) call for proposals for the QCS program I thought the program was wildly overambitious in scope. I also thought that any winning team would quickly fragment into the different team-member groups working in isolation and that it would be impossible to extract any big picture from the outcome. As my fellow team members will attest, I was hesitant to even join the project for these reasons. I now, quite frankly, state that I was wrong. Our team and in fact all the teams pulled together on what really was an immense project and the output, as stated above, was clearly much greater than the sum of its parts. As readers of this document will know, I have been involved in the DoD and Intelligence program in quantum computing since its inception in 1994 (an inception I myself helped to incept). In those nearly 20 years I have witnessed physics experimental groups, physics theory groups, computer science and algorithm groups, complexity theorists groups, and error correction groups all working in, really, mostly in isolation from each other.

Until the QCS program I did not really have even an intuitive sense of how all of this work would someway fit together into the development of a quantum computer. One of the critical successes of the QCS program it developed the tools and more importantly the common language needed for me (and now many others) to see the big picture. To mangle an old metaphor, I feel like a blind man that for 20 years has been groping his way around an elephant, only to find that a cruel trick had been played on me and that I was not blind but just blindfolded. The QCS program has removed that blindfold for me and I’m sure for many others. I have always been an outspoken proponent of the government program to pursue the development of a quantum computer. But I have always had some nagging doubts. Perhaps I had missed something. Perhaps there is some fundamental physical reason or practical technological reason why a large-scale universal quantum computer could never be built — as many distinguished scientists still claim to this day. (See Chapter 15 of Aaronson’s book, for a list of such claims).

Well after participating in the QCS program I have gone from a somewhat hesitant promoter of quantum computing development to now being all in. There is nothing that we have missed. The development of a large-scale universal quantum computer is a formidable technological challenge, and it will likely take tens of years to complete, but there is no chance it is a technological or physical impossibility. To my mind the QCS program results has ruled the latter out — or at least painted it into a very small corner of improbability. The development of a large-scale universal machine is a mathematical certainty and it is just a matter of time before we get there.

Of this I have no longer any doubt.

Acknowledgements: I would like to thank Scott Alexander from Applied Communication Sciences, our team's Principle Investigator, for comments and suggestions and particularly adding the bit about HACKMEM. This blog post was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center contract number D12PC00527. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the U.S. Government.




Monday, February 24, 2014

Who's on First!?

“On the Uncertainty of the Ordering of Nonlocal Wavefunction Collapse when Relativity is Considered,”

by Chris D. Richardson and Jonathan P. Dowling

This preprint on the ArXiv by my former PhD student, Chris Richardson, and me, is not getting enough of the well-deserved publicity that it warrants so here I am to shamelessly promote it. (That and Chris will shortly be looking for a job.) This is really 90% Chris’s work with 10% motivational pep talks from me to him. In a previous blog post, “On the Curious Consistency of Non-Relativistic Quantum Theory with Non-Quantum Relativity Theory,” I blathered on about how odd it was that non-relativistic quantum theory always seemed to be consistent with ordinary relativity theory; even though we have no right to expect that these two theories should be consistent and every right to believe they should flat out contradict each other. This curiousness Nicolas Gisin calls the “tension” between the two theories and Gisin has even done an experiment of the EPR type with a well separated Alice and Bob, but with Bob placed in a moving reference frame (compared to Alice) to try to measure the ‘speed of collapse’ of the two-particle wave function. Gisin and his group conclude that the speed of collapse is some 10,000 times faster than the speed of light (which is consistent with infinitely fast).

This experiment motivated Chris and I to think about a closely related problem; a problem that in fact motivated Gisin’s experiment in the first place. In non-relativistic quantum theory, in an EPR experiment, if Alice makes a measurement on her particle then the state of Bob’s particle is supposed to collapse ‘instantaneously and simultaneously’ to the result anti-correlated to Alice’s measurement (if they share, say, a spin-singlet state). But words like ‘instantaneous’ and — Heaven forbid! — ‘simultaneous’ are heresy in non-quantum relativity theory.

This thought experiment gives rise to a purported paradox. If in one reference frame Alice measures first and collapses Bob’s state there can always be an observer in a different inertial frame who thinks Bob measured first and collapsed Alice’s state. The paradox then may be stated, “Who really collapsed whom first?” This curious swapping of temporal order is the paradox.

Now if I was David Deutsch I would tell you to avoid all this collapse nonsense and to instead embrace the Many Worlds Interpretation of Quantum Mechanics but instead Chris and I decided to push this paradox into a small logical corner where we could beat the heck out of it with a purely Copenhagen Gedanken experimental analysis.

The conclusion of our short paper, now in referee limbo in some journal I’d rather not mention (so as to avoid the once and future lawsuits) is that Naturenot the journal! — deploys a type of quantum-mechanical cloaking device upon the experiment to keep the paradox from arising in the first place.

The paroxysm of paradoxism is swept squarely under a round rug.

The crux of our argument is that Alice and Bob’s measurements cannot be made with infinite precision but are constrained by the Heisenberg uncertainty principle — particularly the notorious energy-time uncertainty principle. Since energy and time are not relativistically invariant quantities, different observers in different reference frames must transform their uncertainty principles accordingly.

Therein lies the rug.

To quote the conclusion of our paper; “The uncertainty in time always outruns the time difference induced by the change in reference frames. Neither Alice nor Bob will ever, with certainty, observe the two measurements swap temporal order.”

Paradox, schmäradox!

The curious consistency of quantum theory and relativity theory hides again….

Monday, February 17, 2014

Boson sampling with photon-added coherent states

Kaushik P. Seshadreesan, Jonathan P. Olson, Keith R. Motes, Peter P. Rohde, Jonathan P. Dowling
Boson sampling is a simple and experimentally viable model for non-universal linear optics quantum computing. Boson sampling has been shown to implement a classically hard algorithm when fed with single photons. This raises the question as to whether there are other quantum states of light that implement similarly computationally complex problems. We consider a class of continuous variable states---photon added coherent states---and demonstrate their computational complexity when evolved using linear optical networks and measured using photodetection. We find that, provided the coherent state amplitudes are upper bounded by an inverse polynomial in the size of the system, the sampling problem remains computationally hard.

See: arXiv:1402.0531