Quantum Computing in the NISQ era and beyond
John Preskill
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena CA 91125, USA

arXiv:1801.00862v3 [quant-ph] 31 Jul 2018

30 July 2018

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in
the near future. Quantum computers with 50-100 qubits may be able to performtasks which surpass the capabilities of today’s classical digital computers, but
noise in quantum gates will limit the size of quantum circuits that can be
executed reliably. NISQ devices will be useful tools for exploring many-bodyquantum physics, and may have other useful applications, but the 100-qubit
quantum computer will not change the world right away — we should regard
it as a significant step toward the more powerful quantum technologies of the
future. Quantum technologists should continue to strive for more accuratequantum gates and, eventually, fully fault-tolerant quantum computing.

1 Introduction
Now is an opportune time for a fruitful discussion among researchers, entrepreneurs, managers, and investors who share an interest in quantum computing. There has been a recent
surge of investment by both large public companies and startup companies, a trend thathas surprised many quantumists working in academia. While we have long recognized thecommercial potential of quantum technology, this ramping up of industrial activity has
happened sooner and more suddenly than most of us expected.
In this article I assess the current status and future potential of quantum computing.
Because quantum computing technology is so different from the information technology
we use now, we have only a very limited ability to glimpse its future applications, orto project when these applications will come to fruition. While this uncertainty fuels
optimism, our optimism should be tempered with caution. We may feel confident thatquantum technology will have a substantial impact on society in the decades ahead, but
we cannot be nearly so confident about the commercial potential of quantum technology inthe near term, say the next five to ten years. That is the main message I hope to convey.
With that said, I’m sure that vigorous discussion among all the interested parties can help
light the way toward future progress.

2 Opportunities at the entanglement frontier
I am a theoretical physicist with a background in particle physics and cosmology, but
for more than 20 years much of my research effort has been directed toward quantum
information science. I’m drawn to this field because I feel we are now in the early stages
John Preskill: preskill@caltech.edu, Based on a Keynote Address at Quantum Computing for Business, 5 December 2017.

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1

of exploring a new frontier of the physical sciences, what we might call the complexity
frontier or the entanglement frontier. This new frontier, while different than the frontierof particle physics or cosmology, is very fundamental and exciting. Now, for the first timein human history, we are acquiring and perfecting the tools to build and precisely control
very complex, highly entangled quantum states of many particles, states so complex that
we can’t simulate them with our best digital computers or characterize them well using
existing theoretical tools. This emerging capability will open the door to new discoveries.
For a physicist like me, what is really exciting about quantum computing is that wehave good reason to believe that a quantum computer would be able to efficiently simulate
any process that occurs in Nature. We don’t think that’s not true for classical (i.e., nonquantum) digital computers, which (as far as we know) can’t simulate highly entangled
quantum systems. With quantum computers we should be able to probe more deeply into
the properties of complex molecules and exotic materials, and also to explore fundamental
physics in new ways, for example by simulating the properties of elementary particles, orthe quantum behavior of a black hole, or the evolution of the universe right after the bigbang.
Our confidence that exploration of the entanglement frontier will be rewarding rests
largely on two principles: (1) quantum complexity (our basis for thinking that quantum
computing is powerful), and (2) quantum error correction (our basis for thinking thatquantum computers are scalable to large devices solving hard problems). Underlying bothof these principles is the idea of quantum entanglement. Entanglement is the word we use
for the characteristic correlations among the parts of a quantum system, which are quite
different from the correlations we encounter in everyday life. To understand the concept ofentanglement, imagine a system with many parts, for example a book which is 100 pages
long. For an ordinary classical 100-page book, every time you read another page you learn
another 1% of the content of the book, and after you have read all of the pages one by
one you know everything that’s in the book. But now suppose instead that it’s a quantum
book, where the pages are very highly entangled with one another. Then when you lookat the pages one at a time you see only random gibberish, and after you have read all thepages one by one you know very little about the content of the book. That’s because theinformation in the quantum book is not imprinted on the individual pages; it is encoded
almost entirely in how the pages are correlated with one another. If you want to read thebook you have to make a collective observation on many pages at once. That’s quantum
entanglement, the essential feature making information carried by quantum systems very
different from information processed by ordinary digital computers.

3 The potential of quantum computing
3.1 Why we think quantum computing is powerful
An ordinary computer processes bits, where each bit could be, say, a switch which is either
one or off. But to build highly complex entangled quantum systems, the fundamental
information-carrying components of a quantum computer must be quantum bits, what wecall qubits. A qubit can be realized physically in many different ways. It can be carried
by a single atom, or a single electron, or a single photon (a particle of light). Or a qubit
can be carried by a more complicated system, like a very cold superconducting electricalcircuit in which many electrons are moving.
When we speak of quantum complexity, what springs to mind is the staggering complexity of using ordinary classical data to describe highly entangled quantum states ofAccepted in Quantum 2018-07-30, click title to verify

2many qubits. Giving a complete description of all the correlations among just a few hundred qubits may require more bits than the number of atoms in the visible universe. It
will never be possible, even in principle, to write that description down, or to faithfully
describe the processing of a few hundred qubits using classical language.
This apparently extravagant complexity of the quantum world, though highly suggestive, does not by itself ensure that quantum computers are more powerful than classicalones. But we have at least three good reasons for thinking that quantum computers havecapabilities surpassing what classical computers can do.
(1) Quantum algorithms for classically intractable problems. First, we know of problems
that are believed to be hard for classical computers, but for which quantum algorithms have been discovered that could solve these problems easily. The best known
example is the problem of finding the prime factors of a large composite integer [1].
We believe factoring is hard because many smart people have tried for many decades
to find better factoring algorithms and haven’t succeeded. Perhaps a fast classicalfactoring algorithm will be discovered in the future, but that would be a big surprise.
(2) Complexity theory arguments. The theoretical computer scientists have provided arguments, based on complexity theory, showing (under reasonable assumptions) thatquantum states which are easy to prepare with a quantum computer have superclassical properties; specifically, if we measure all the qubits in such a state we aresampling from a correlated probability distribution that can’t be sampled from byany efficient classical means [2, 3].
(3) No known classical algorithm can simulate a quantum computer. But perhaps themost persuasive argument we have that quantum computing is powerful is simply
that we don’t know how to simulate a quantum computer using a digital computer;that remains true even after many decades of effort by physicists to find better ways
to simulate quantum systems.
It’s a remarkable claim — one of the most amazing ideas I’ve encountered in myscientific life — that there is a distinction between problems that are classically hard andproblems that are quantumly hard. And it is a compelling challenge to understand better
what problems are classically hard but quantumly easy [4, 5]. We should recognize inparticular that the power of a quantum computer is not unlimited. We don’t expect, forexample, that a quantum computer will be able to solve efficiently the hard instances of NPhard problems like the traveling salesman problem. For such hard combinatorial search
problems we probably can’t do much better than exhaustively searching for a solution.
Quantum computers can speed up exhaustive search [6], but only modestly [7], so NPhard problems are likely to be quantumly hard as well as classically hard.
For a physicist seeking problems which are classically hard and quantumly easy, thenatural place to look is the task of simulating a many-particle quantum system. As two
great physicists, Bob Laughlin and David Pines, put it some years ago [8], we have a “theory of everything that is relevant to ordinary life.” We have high confidence this theory
is correct, and we can write down the equations precisely — they are the equations that
describe how atomic nuclei and electrons interact electromagnetically. But we can’t solve
those equations. And so as Laughlin and Pines put it: “We have a theory of everything only
to discover that it has revealed exactly nothing about many things of great importance.”
Those things of importance they envisaged are the situations in the quantum world whereentanglement has profound consequences. Dramatizing the futility of the task they pro-

Accepted in Quantum 2018-07-30, click title to verify3claimed: “No computer existing, or that will ever exist, can break this barrier” of solving
the equations describing many entangled particles.
But in fact, years before Laughlin and Pines wrote these words, the physicist Richard
Feynman had articulated a rebuttal [9]. As Feynman put it: “Nature isn’t classical dammit,
and if you want to make a simulation of Nature you better make it quantum mechanical, andby golly it’s a wonderful problem because it doesn’t look so easy.” Feynman had envisioned
using a quantum computer to solve the quantum physics problems that physicists andchemists had failed to solve using digital computers. Laughlin and Pines knew well that
Feynman had made this proposal years earlier, but had dismissed his idea as impractical.
Now, some 35 years after Feynman’s proposal, we’re just beginning to reach the stage
where quantum computers can provide useful solutions to hard quantum problems.

3.2 Why quantum computing is hard
So why is it taking so long? What is it about quantum computing that’s so difficult? The
core of the problem stems from a fundamental feature of the quantum world — that wecannot observe a quantum system without producing an uncontrollable disturbance in thesystem. That means that if we want to use a quantum system to store and reliably process
information, then we need to keep that system nearly perfectly isolated from the outside
world. At the same time, though, we want the qubits to strongly interact with one anotherso we can process the information; we also need to be able to control the system from theoutside, and eventually read out the qubits so we can find the result of our computation.
It is very challenging to build a quantum system that satisfies all of these desiderata. It
has taken many years of development in materials and control and fabrication to get where
we are now.
Eventually we expect to be able to protect quantum systems and scale up quantum
computers using the principle of quantum error correction [10]. The essential idea of
quantum error correction is that if we want to protect a quantum system from damage
then we should encode it in a very highly entangled state; like that 100-page book I
described earlier, this entangled state has the property that the environment, interactingwith parts of the system one at a time, is unable to glimpse the encoded information andtherefore can’t damage it. Furthermore, we’ve understood in principle how to process
quantum information which is encoded in a highly entangled state. Unfortunately, there
is a significant overhead cost for doing quantum error correction — writing the protected
quantum information into a highly entangled book requires many additional physical qubits— so reliable quantum computers using quantum error correction are not likely to beavailable very soon.

4 The NISQ era unfolds
4.1 The 50-qubit barrier
Even with fault-tolerant quantum computing still a rather distant dream, we are now
entering a pivotal new era in quantum technology. For this talk, I needed a name to describe
this impending new era, so I made up a word: NISQ. This stands for Noisy IntermediateScale Quantum. Here “intermediate scale” refers to the size of quantum computers which
will be available in the next few years, with a number of qubits ranging from 50 to a

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4few hundred.1 50 qubits is a significant milestone, because that’s beyond what can be
simulated by brute force using the most powerful existing digital supercomputers.2 “Noisy”
emphasizes that we’ll have imperfect control over those qubits; the noise will place serious
limitations on what quantum devices can achieve in the near term.
Physicists are excited about this NISQ technology, which gives us new tools for exploring the physics of many entangled particles. It might also have useful applications ofinterest to the business community, but we’re not sure about that. We shouldn’t expect
NISQ is to change the world by itself; instead it should be regarded as a step toward more
powerful quantum technologies we’ll develop in the future. I do think that quantum computers will have transformative effects on society eventually, but these may still be decades
away. We’re just not sure how long it’s going to take.

4.2 Qubit “quality”
I’ve emphasized the number of qubits as a measure of how difficult it is to do the simulationof a quantum computer on a classical device, but the number of qubits isn’t the only thing
we care about. We also care about the quality of the qubits, and in particular the accuracy
with which we can perform quantum gates — well-controlled entangling operations acting
on pairs of qubits. With the best hardware we have now for controlling trapped ions [14] orsuperconducting circuits [15], the error rate per gate for two-qubit gates is above the .1%
level (and often much worse). Furthermore, we don’t yet know whether error rates that low
can be maintained in larger devices with many qubits; perhaps we’ll find out soon. Naively,then, and as I’ll say later this might be too naive, with these noisy devices we don’t expect
to be able to execute a circuit that contains many more than about 1000 gates — that is,1000 fundamental two-qubit operations — because the noise will overwhelm the signal ina circuit much larger than that. That limitation on circuit size imposes a ceiling on thecomputational power of NISQ technology. Eventually we’ll do better, using quantum error
correction to scale up to larger circuits. But as I’ve already emphasized, since quantum
error correction imposes a heavy overhead cost in number of qubits and number of gates,
scaling up using quantum error correction is a more distant goal. When I speak of the
NISQ era, I’m imagining quantum computers with noisy gates unprotected by quantum
error correction.
There are other things we care about, too, aside from the number of qubits and thegate error rate. The time it takes to execute a single gate is also important for setting
the time scale needed for a quantum computer to solve a problem, and it is noteworthy
that superconducting circuits are about a thousand times faster than ion trap quantum
processors. We need to be able to prepare and measure qubits accurately; currently themeasurement error probability is about 1% for superconducting qubits, and much better
than that for trapped ions. We care about the connectivity among the qubits — forwhich pairs of qubits in our device can we perform an accurate two-qubit gate? It is also
important to know how reliably we can fabricate qubits — if we attempt to construct a
many-qubit device, how many of the qubits actually perform well enough to be useful?
These considerations and others, too, should be kept in mind when we make comparisonsacross different quantum computing platforms.
172-qubit and 50-qubit devices, based on superconducting circuits, have been announced recently byGoogle and IBM, respectively.
2

This “milestone” is actually a bit fuzzy, in several respects. In particular, the resources needed in aclassical simulation depend on the depth (number of time steps) of the quantum circuit, as well as on thenumber of qubits [11, 12, 13].

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55 What I won’t say much about
Before moving on to the main focus of this article, I want to include a disclaimer to avoid
misunderstanding. There are some very interesting and important aspects of quantum
technology that I’m not really going to talk about, but I’ll mention them so you’ll know
I’m not going to talk about them.
(1) Quantum-resistant cryptography. We foresee disruptive effects of quantum computerson how we protect our privacy — the public key cryptosystems that are in widespread
use today will become obsolete in the years ahead because they can be easily broken
by sufficiently powerful quantum computers. We should be thinking now about how
we’re going to protect our privacy in the future. One possible way is to replace ourexisting cryptosystems with new ones which we’re confident are resistant to attacks
by quantum computers [16].
(2) Quantum key distribution, quantum networks, and quantum repeaters. We might also
use traveling qubits — most likely photons — to create shared keys that can be used
for encryption and decryption. Quantum key distribution exploits the principle thatyou cannot eavesdrop on quantum communication without producing a detectabledisturbance [17]. But we don’t yet have the technology to distribute quantum entanglement, and hence secret key, around the world; that’s another interesting technological challenge [18]. It is also notable that, aside from applications to key exchange,a global quantum network might be used for other purposes, like sharing information
among quantum devices.
(3) Quantum randomness expansion. Unlike deterministic classical devices, quantum
devices can generate intrinsic randomness. Remarkably, this feature of quantum
physics can be exploited to expand a short random seed to a much longer string ofcertifiably random bits, even if one distrusts the equipment used for this purpose —it suffices to assume that spacelike separated parties are unable to communicate with
one another [19], or to make suitable assumptions about (quantum) computational
hardness [20]. Certifiable randomness has many potential applications, for example
to secure communication protocols, unbiased statistical sampling, and Monte Carlosimulations.
(4) Quantum sensing. Quantum technology has advantages for some kinds of sensing.
Quantum systems can sense weak forces with better sensitivity and higher spatial
resolution than other sensing technologies [21]; thus quantum sensing could haverelatively near-term high-impact applications, to medicine for example.
Quantum communication, networking, randomness expansion, and sensing cannot becleanly separated from quantum computing, because similar technological challenges are
faced by all these quantum-information goals. Nevertheless, I’ll give short shrift to these
topics here.

6 Quantum speedups?
What I would like to focus on instead is whether quantum computers will have widelyused applications, particularly in the relatively near term. The main question is: When
will quantum computers be able to solve problems we care about faster than classicalcomputers, and for what problems?
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6At least in the near term quantum computers are likely to be special purpose devices,which most users will access via the cloud. When we speak of a quantum speedup, we
typically mean that the quantum computer solves the problem faster than competingclassical computers using the best available hardware and running the best algorithm which
performs the same task. (Arguably, though, quantum technology might be preferred evenif classical supercomputers run faster, if for example the quantum hardware has lower
cost and lower power consumption.) In any case, we should recognize that the powerof classical computers will continue to increase, with exascale systems (surpassing 1018
FLOPS) expected to be available in a few years. Quantum computers are striving to catch
up with a moving target, as both classical hardware and classical algorithms continually
improve.
A few years ago I spoke enthusiastically about quantum supremacy as an impending
milestone for human civilization [22]. I suggested this term as a way to characterize computational tasks performable by quantum devices, where one could argue persuasively thatno existing (or easily foreseeable) classical device could perform the same task, disregarding whether the task is useful in any other respect. I was trying to emphasize that now
is a very privileged time in the coarse-grained history of technology on our planet, and I
don’t regret doing so. But from a commercial perspective, obviously we should pay attention to whether the task is useful! Quantum supremacy is a worthy goal, notable forentrepreneurs and investors not so much because of its intrinsic importance but rather asa sign of progress toward more valuable applications further down the road.
We should also bear in mind that, because of the imperfect performance of NISQ
technology, it may be hard to validate that a quantum computer is really giving the rightanswer. That’s particularly true for the quantum simulation problems physicists are excitedabout. So it’s important for researchers to continue seeking better methods for verifying
the output of a quantum computer.

6.1 Quantum optimizers
I have already emphasized that we don’t expect quantum computers to be able to solve
efficiently worst-case instances of NP-hard problems like combinatorial optimization problems; still, it’s conceivable (though not guaranteed) that quantum devices will be able tofind better approximate solutions or find such approximate solutions faster. For example,
we might formulate m constraints on n bits, and seek an n-bit string which solves as manyof the m constraints as possible. We may say we have solved this optimization problem
exactly if we can find the maximal number k of constraints that can be satisfied simultaneously, and that we have an approximate solution if we can guarantee that the maximal
number of satisfiable constraints is at least k 0 for some k 0 < k; the approximation ratio
k 0 /k is a measure of the quality of our approximate solution.
For some optimization problems, even finding an approximation solution is NP-hard
if the approximation ratio is sufficiently close to one [23], and in those cases we don’t
expect a quantum computer to be able to find the approximate solution efficiently forhard instances of the problem. But for many problems there is a big gap between theapproximation achieved by the best classical algorithm we currently know and the barrierof NP-hardness. So it would not be shocking to discover that quantum computers have an
advantage over classical ones for the task of finding approximate solutions, an advantagethat some users might find quite valuable.
Whether that quantum advantage really exists is currently an open question, but we willsoon have the opportunity to explore the question experimentally using quantum hardware.
Of course, even if there is such a quantum advantage for approximate optimization, NISQ
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7technology might be inadequate for demonstrating the advantage; still, it will be fun totry it and see how well it works.
The emerging paradigm for solving optimization problems using near-term quantum
technology is a kind of hybrid quantum-classical algorithm. In this scheme, we use aquantum processor to prepare an n-qubit state, then measure all the qubits and process
the measurement outcomes using a classical optimizer; this classical optimizer instructs
the quantum processor to alter slightly how the n-qubit state is prepared. This cycle isrepeated many times until it converges to a quantum state from which the approximate
solution can be extracted. When applied to classical combinatorial optimization problems,
this procedure goes by the name Quantum Approximate Optimization Algorithm (QAOA)[24]. But it can also be applied to quantum problems, like finding low-energy states ofmany-particle quantum systems (large molecules, for example). When applied to quantum
problems this hybrid quantum-classical procedure goes by the name Variational Quantum
Eigensolver (VQE) [25].
Will NISQ technology running QAOA or VQE be able to outperform classical algorithms that find approximate solutions to the same problems? Nobody knows, but we’regoing to try it and see how well we can do. It’s an ambitious goal, because the classicalmethods we use to solve these problems are well honed after decades of development. On
the other hand, even if early-generation NISQ devices are not yet ready to compete withthe best classical computers, experimental results might encourage our hopes that QAOA
and/or VQE can surpass classical methods in the future, and so spur further technologicalimprovements.

6.2 How quantum testbeds might help
The history of classical computing teaches us that when hardware becomes available that
stimulates and accelerates the development of new algorithms. There are many examples of heuristics that were discovered experimentally, which worked better than theorists
could initially explain. We can anticipate that the same thing will happen with quantum
computers.
For example, theorists eventually explained why the simplex method for linear programming works well in practice [26], but only long after it had been found to be usefulexperimentally. A current example is deep learning [27]; we lack a good theoretical explanation for why it works as well as it does. In the quantum case, too, experiments may
validate the performance of heuristic algorithms, where we don’t understand why theywork. That might happen in particular for the quantum optimization algorithms we’replanning to test.
But I emphasize again that the imperfect gates in NISQ systems will severely limit
their computational power, because circuits with many gates will be too noisy to give
useful results. Near-term experiments will explore what we can do with of order 100
qubits, and with circuit depth (number of time steps) less than 100, maybe much less. A
vibrant discussion between quantum algorithm designers and application users might point
those experiments in promising directions; I hope that this article can help to facilitate
and inspire that discussion.

6.3 Quantum annealing
I have emphasized 50 to 100 qubits as a coming milestone for quantum technology, but infact we have a 2000-qubit quantum device now, the D-Wave 2000Q machine. This machine
is not a circuit-based quantum computer. Rather it is what we call a quantum annealer; it
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8solves optimization problems using a different method than execution of a quantum circuit,and it often solves these problems successfully.
But as of now we don’t have a convincing theoretical argument or persuasive experimental evidence indicating that quantum annealers can really speed up the time to solution
compared to the best classical hardware running the best algorithms for the same problems[28, 29]. The situation is a bit nuanced. The quantum annealer is the noisy version (withrather poor quality qubits, in the case of the D-Wave machine) of what we call adiabatic
quantum computing [30]. We actually do have, for the case of noiseless qubits, a theoreticalargument showing that adiabatic quantum computing is as powerful as circuit-based quantum computing [31]. However, the argument demonstrating the equivalence of adiabaticand circuit-based quantum computation applies only if the adiabatic method is executed ina form that requires a high overhead cost in additional physical qubits, which is quite different from the way quantum annealers are used today. In any event, this formal argument
applies only to noiseless qubits, and (in contrast to circuit-based quantum computing),
we don’t have a good theoretical argument attesting that quantum annealers are scalable.
That is, for the case of noisy qubits we don’t know how to show that quantum annealers
will continue to work successfully as the problem size increases, even if we are willing toaccept a sizable overhead cost to enforce robustness against noise.
So far quantum annealers have been applied mostly to cases where the annealing is
stoquastic — that means it might be relatively easy for a classical computer to simulate
what the quantum annealer is doing [32, 33]. What’s coming soon are non-stoquastic
quantum annealers, which may have greater potential for achieving speedups over what
the best classical algorithms can do.
Since theorists have not settled whether quantum annealing is powerful, further experiments are needed. Experiments with quantum annealers over the next few years are likely
to be quite informative. In particular, aside from the applications of quantum annealers toclassical optimization problems, applications to quantum simulation problems should also
be explored [34].

6.4 Noise-resilient quantum circuits
I’ve emphasized that eventually we are going to use quantum error correction to extend
the size of the computations we can execute reliably using noisy quantum computers. But,
because of the high overhead cost of quantum error correction, NISQ devices will not makeuse of it in the near term. Nevertheless, methods for mitigating the effects of noise may
be an important consideration during the NISQ era.
Naively one might expect that, for a generic circuit with G gates, a single fault anywherein the circuit could cause the computation to fail; if so, then we cannot execute the circuit
reliably if the error rate per gate is much larger than 1/G. However, at least for someproblems and algorithms, that conclusion might be too pessimistic.
In particular, for some of the quantum simulation algorithms of interest to physicists,
there are only a relatively small number of circuit locations where a faulty gate can cause
the computation to fail badly. This feature could arise because the circuit has low depth,
and we can tolerate a constant probability of error per qubit in the final measurement outcome, or because the damage caused by an error occurring early in the circuit decays away
by a later time. Such noise resilience is characteristic of some tensor network constructions which can be invoked for solving quantum optimization problems using Variational
Quantum Eigensolvers [35, 36].
There is a substantial opportunity for experimentalists and theorists, working together
over the next few years, to find better ways of making quantum circuits noise resilient,Accepted in Quantum 2018-07-30, click title to verify

9and so extend the computational reach of NISQ technology. We should be wary, though,of a potential tradeoff — making a quantum circuit more noise resilient may also make iteasier to simulate classically.

6.5 Quantum deep learning
Machine learning is transforming technology and having a big impact on science as well, so
it is natural to wonder about the potential of combining machine learning with quantum
technology. There are a variety of different notions of “quantum machine learning.” Muchof the literature on the subject builds on quantum algorithms that speed up linear algebra
and related tasks [37, 38], and I’ll address such applications in the ensuing subsections.
First, though, I’ll comment on the potential of quantum deep learning [39]. Deep learning itself has become a broad subject [27], but to be concrete let us contemplate a restricted
Boltzmann machine. This may be regarded as a spin system in thermal equilibrium at a
low but nonzero temperature, with many hidden layers of spins separating the input andoutput. (The word “restricted” means that there are no couplings among spins within agiven layer; instead only spins in successive layers are coupled.) The system may havemillions of coupling parameters, which are optimized during a training phase to achievea desired joint probability distribution for input and output. The training may be supervised, for example the network might be taught to recognize a labeled set of photos, or
unsupervised, in which case the goal is to identify patterns in an unlabeled training set ofdata. The quantum analog of such a machine could have a similar structure and function,
but where the spins are qubits rather than classical bits. It may be that quantum deeplearning networks have advantages over classical ones; for example they might be easier totrain for some purposes. But we don’t really know — it’s another opportunity to try it
and see how well it works.
One possible reason for being hopeful about the potential of quantum machine learning
rests on the concept known as QRAM — quantum random access memory. By QRAM wemean representing a large amount of classical data very succinctly, by encoding a vector
with N components in just log N qubits. But typical proposals for quantum machinelearning applications are confounded by severe input/output bottlenecks. For applications
to large classical data sets one should take into account the cost of encoding the input
into QRAM, which can nullify the potential advantages. Furthermore, the output from aquantum network is itself a quantum state; to get useful information we need to measure
the state, and we can acquire no more than log N bits of classical information in a singleshot measurement of log N qubits.
These bottlenecks arise when we try to train a quantum network to learn about correlations in classical data. Perhaps it is more natural to think about quantum machinelearning in a setting where both the input and the output are quantum states. Hence wemight expect a quantum network to have advantages over a classical one when used, forexample, to control a complex quantum system. More broadly, it seems plausible thatquantum deep learning would be more powerful than its classical counterpart when trying
to learn probability distributions for which quantum entanglement has a significant role.
That is, quantum deep learning networks might be very well suited for quantum tasks, but
for applications of deep learning that are widely pursued today it is unclear why quantum
networks would have an advantage.
On the encouraging side, it’s worth noting that a quantum deep learning machine won’t
necessarily need to be a general purpose circuit-based quantum computer. It might be a
special purpose device — perhaps a quantum annealer, if not too noisy.

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106.6 Quantum matrix inversion
QRAM has further implications for quantum algorithms. In particular, the task of matrix
inversion admits an exponential quantum speedup, which could have many applications.
The algorithm we call HHL [40] (after the three authors who first discussed it nine yearsago) takes as input a succinct representation of a very large N ×N matrix A, which has
to be sufficiently sparse and well conditioned, and also as input an N -component vector|bi, encoded in QRAM as a quantum state of log N qubits. The output of the quantum
algorithm is (with a small error) the quantum state |A−1 bi, the result of applying theinverse of the matrix A to the input vector |bi. This quantum algorithm runs in timeO(log N ), an exponential speedup relative to classical matrix inversion (for fixed error andfor fixed sparsity and condition number of A).
However, aside from the caveat that the matrix A must be sparse and well conditioned,
note that both the input vector |bi and the output vector |A−1 bi are quantum states. Once
we have the output state, we can perform measurements to learn features of the output
vector A−1 b, and by repeating the scheme many times we can learn about the output vector
in greater detail. But if we are trying to apply this matrix inversion algorithm to classicalinput data, we need to take into account the potential cost of loading that classical data
into QRAM, which could nullify the exponential speedup. Alternatively, loading classicaldata into QRAM might be avoided by computing |bi on a quantum computer rather thanentering it from a database.
We do have good reason to believe this quantum matrix inversion algorithm is powerful,
because it solves what we call a BQP-complete problem. That is, any problem that can besolved efficiently with a quantum computer can be encoded as an instance of this matrix
inversion problem. And a number of potentially useful applications have been suggested.
For example, we can use this matrix inversion algorithm to find approximate solutions toclassical linear field equations [41, 42, 43] (after discretizing the field on a spatial lattice,
and applying a suitable preconditioner). This procedure could be used, say, to solve theequations of electromagnetism in a complex three-dimensional geometry, for the purposeof optimizing the performance of an antenna.
I expect HHL to have high-impact applications eventually, but it is not likely to be feasible in the NISQ era. The algorithm may just be too expensive to be executed successfully
by a quantum computer which doesn’t use error correction.

6.7 Quantum recommendation systems
Another recently proposed quantum algorithm [44] provides an exponential speedup overthe best currently known classical algorithm for a useful task: making high-value recommendations. The goal is to recommend a product to a customer that the customer willprobably like, based on limited knowledge of the preferences of that customer and othercustomers.
To formulate a simplified version of the problem, consider m customers and n products,
and let P be a binary m×n preference matrix such that Pai = 1 if customer a likes producti and Pai = 0 if customer a dislikes product i. In practice we might have m ≈ 108 users andn ≈ 106 products, but the rank k of the matrix is relatively small, perhaps k ≈ 100. This
small rank indicates that there are only a limited number of customer types; therefore,
once we know a few of the preferences of a new customer, we can accurately recommend
other products that customer will like.
An algorithm for recommendation systems has two stages. In the first stage, performed
offline, a low rank approximation to the preference matrix P is constructed. In the second

Accepted in Quantum 2018-07-30, click title to verify

11stage, performed online, a new customer reveals some preferences, and the system outputs a
recommendation which that customer will like with high probability. It is this second online
stage that admits a quantum speedup. The quantum runtime is O(poly(k)polylog(mn)),while the best classical algorithm known requires time poly(mn) to return a high-value
recommendation. Crucially, the quantum algorithm, in contrast to the classical one, does
not attempt to reconstruct the full recommendation matrix, which would take too long.
Instead it samples efficiently from the low-rank approximation to P .
This is a significant quantum speedup for a real-world application of machine learning,
encouraging the hope that other such speedups will be discovered. However, we don’t currently have a convincing theoretical argument indicating that the task performed by quantum recommendation systems (returning a high-quality recommendation in polylog(mn)time) is impossible classically.3 Formulating such an argument would strengthen our confidence that quantum computers will have a significant impact on machine learning. Heretoo, though, the algorithm is probably too costly to be convincingly validated during the
NISQ era.

6.8 Quantum semidefinite programming
Semidefinite programming is the task of optimizing a linear function subject to matrix
inequality constraints. Specifically, given as input the m+1 N ×N Hermitian matrices{C, A1 , A2 , . . . , Am } and real numbers {b1 , b2 , . . . , bm }, the problem is to find a positive semidefinite N × N matrix X that maximizes tr (CX) subject to the m constraintstr (Ai X) ≤ bi . Convex optimization problems of this type have widespread applications,and can be solved classically in poly(m, N ) time.
A recently discovered quantum algorithm [46, 47] finds an approximate solution to
the problem in time polylog(N ), an exponential speedup. In the quantum algorithm, theoutput is a quantum state, a density operator ρ which approximates the optimal matrix
X. This quantum state can then be measured to extract information about X, and theprocedure can be repeated many times to learn about X is more detail.
However, in the analysis of the quantum algorithm one assumes the availability of
an initial quantum state of a particular form, the thermal Gibbs state associated witha Hamiltonian which is a linear combination of the input matrices for the semidefinite
program. Whether the exponential speedup can actually be realized, then, depends onwhether this Gibbs state can be prepared efficiently using a quantum computer.
For semidefinite programs with the right structure, the efficient preparation of the
Gibbs state is possible, and therefore the exponential speedup really is feasible. In particular, the quantum algorithm achieves an exponential speedup if all the input matrices
for the semidefinite program have low rank, or if the corresponding Hamiltonian system
thermalizes rapidly. It is not yet clear to what extent these properties apply to semidefinite programs of practical interest; clarifying that issue is an important goal for further
research.
It is intriguing that the crucial step in the quantum algorithm is the preparation ofa thermal state at a nonzero temperature. That may suggest that the algorithm hassome intrinsic robustness against thermal noise, or that the thermalization stage of thealgorithm could be executed by quantum annealing. Conceivably, then, a quantum solver
3

Note added: After this paper was submitted, Tang [45] exhibited a “quantum-inspired” classical algorithm which returns a high-value recommendation in runtime O(poly(k)polylog(mn)); thus the quantum
algorithm proposed in [44] does not in fact achieve an exponential speedup compared to the best classicalalgorithm performing the same task.

Accepted in Quantum 2018-07-30, click title to verify

12for semidefinite programs might be within reach of NISQ technology.

6.9 Quantum simulation
As I have already emphasized, we expect quantum computers to be very well suited forstudying the properties of highly entangled systems of many particles. As Laughlin andPines [8] emphasized in the statement I quoted earlier, we think that simulating such“strongly correlated” matter is a difficult computational problem, because many very good
physicists have tried to solve it for decades and have not succeeded. Thus, as Feynman
argued [9], the simulation of highly entangled quantum matter is the natural arena where
quantum computers seem to have a clear advantage over classical ones.
In the long term, quantum simulation using quantum computers has great potential
to wield a broad influence on the world [48, 49, 50]. Advances in quantum chemistry,
facilitated by quantum computing, might precipitate the design of new pharmaceuticals,
and new catalysts that improve the efficiency of nitrogen fixation or carbon capture. New
materials arising from quantum simulations might lead to more effective power transmission
or improved collection of solar energy. We probably lack the imagination to anticipate themost transformative discoveries which will flow from more powerful quantum simulators.
But this promise is not so likely to be fulfilled in the NISQ era, because quantum algorithms
for accurately simulating large molecules and exotic materials may be too expensive to
succeed without error correction.
Classical computers are especially bad at simulating quantum dynamics — that is, predicting how a highly-entangled quantum state will change with time. Quantum computers
have a big advantage for that task, and physicists hope to learn interesting things about
quantum dynamics using NISQ technology in the relatively near term. It is instructive torecall that the theory of classical chaos (the extreme sensitivity to initial conditions in classical dynamical systems, which accounts for our inability to predict the weather more than
two weeks out) advanced rapidly in the 60s and 70s after it became possible to simulate
chaotic dynamical systems using classical computers. We may anticipate that the emerging
ability to simulate chaotic quantum systems (those in which entanglement spreads very
rapidly) will promote advances in our understanding of quantum chaos. Valuable insights
might already be gleaned using noisy devices with of order 100 qubits.

6.10 Digital vs. analog quantum simulation
Having emphasized the potential value of simulating quantum dynamics, I should comment
about the distinction between analog and digital quantum simulation. When we speak of ananalog quantum simulator we mean a system with many qubits whose dynamics resembles
the dynamics of a model system we are trying to study and understand. In contrast, a
digital quantum simulator is a gate-based universal quantum computer which can be used
to simulate any physical system of interest when suitably programmed, and can also be
used for other purposes.
Analog quantum simulation has been a very vibrant area of research for the past 15
years, while digital quantum simulation with general purpose circuit-based quantum computers is just now getting started. But some of the same experimental platforms, forexample trapped ions and superconducting circuits, can be used for both purposes. Analog quantum simulators have been getting notably more sophisticated, and are alreadybeing employed to study quantum dynamics in regimes which may be beyond the reach ofclassical simulators [51, 52]. But analog quantum simulators are hampered by imperfectcontrol — the actual quantum system in the lab only crudely approximates the model
Accepted in Quantum 2018-07-30, click title to verify

13system of interest. For that reason, analog simulators are best suited for studying features that physicists call universal, properties which are relatively robust with respect tointroducing small sources of error. A major challenge for research using analog quantum
simulators is identifying accessible properties of quantum systems which are robust withrespect to error, yet are also hard to simulate classically.
We can anticipate that analog quantum simulators will eventually become obsolete.
Because they are hard to control, they will be surpassed some day by digital quantum
simulators, which can be firmly controlled using quantum error correction. But becauseof the hefty overhead cost of quantum error correction, the reign of the analog quantum
simulator may persist for many years. Therefore, when seeking near-term applicationsof quantum technology, we should not overlook the potential power of analog quantum
simulators.

6.11 Quantum games
Advances in classical computing launched a new world of digital games, touching the livesof millions and generating billions in revenue. Could quantum computers do the same?
Physicists often say that the quantum world is counter-intuitive because it is so foreign
to ordinary experience. That’s true now, but might it be different in the future? Perhaps
kids who grow up playing quantum games will acquire a visceral understanding of quantum
phenomena that our generation lacks. Furthermore, quantum games could open a niche forquantum machine learning methods, which might seize the opportunity to improve gameplay in situations where quantum entanglement has an essential role.

7 The daunting climb to scalability
I’ve already emphasized repeatedly that it will probably be a long time before we havefault-tolerant quantum computers solving hard problems. Nevertheless, we can make significant progress in the near term by developing better methods and hardware for implementing quantum error correction, guided by relatively small-scale experiments withquantum error-correcting codes. We expect that, in the next few years, enhanced control
of an error-corrected qubit will be demonstrated convincingly in the lab for the first time.
Further near-term experiments exploring quantum error correction are bound to advanceour understanding of noise in quantum devices, and to suggest more effective techniques
for noise mitigation [53, 54] and device validation [55].
Nevertheless, solving really hard problems (like factoring numbers which are thousandsof bits long) using fault-tolerant quantum computing is not likely to happen for a while,
because of the large number of physical qubits needed. To run algorithms involving thousands of protected qubits we’ll need a number of physical qubits which is in the millions,or more [56]. That’s a very large leap from where we will be for the next few years, with oforder a hundred qubits. Crossing the “quantum chasm,” from hundreds of physical qubits
to millions of physical qubits, is going to take some time, but we’ll get there eventually.
Throughout the NISQ era it will remain important to strive for lower gate error rates inthe various quantum platforms. With more accurate quantum gates, quantum computers
operating without error correction will be able to go further by executing larger circuits.
Better gates will also lower the overhead cost of doing quantum error correction when
we eventually move on to fault-tolerant quantum computing in the future. (Microsoft’s
current program [57], focused on advancing topological quantum computing, is foundedon the notion that much lower gate error rates will pay off handsomely sometime in the

Accepted in Quantum 2018-07-30, click title to verify

14

future.)It is important to realize that we will need significant advances — in basic science aswell as in systems engineering — to attain fully scalable fault-tolerant quantum computers. Therefore, the challenge of scalability poses compelling problems for scientists andengineers. Because we still have so far to go, new insights, developments, and innovations
have the potential to substantially alter the long-term outlook.

8 SummaryThis article is the result of an uneasy compromise. I hope the content provides a useful
overview for technically informed readers, but I’ve tried to minimize jargon and technicaldetails which could be distracting or off-putting for a broader audience. I have not attempted to compile a comprehensive list of references; instead I have referred to just a few
papers that amplify or clarify points I wanted to make, or that might provide instructive
further reading. I apologize to many colleagues whose highly valuable contributions are
not properly acknowledged here.
Let me now summarize the main points I’ve tried to convey.
Now is a privileged time in the history of science and technology, as we are witnessing the
opening of the NISQ era (where NISQ = noisy intermediate-scale quantum). We’llsoon have the opportunity to experiment with NISQ technology to see what it cando. Perhaps NISQ will allow us to speed up the time to solution for problems ofbroad interest in the near future, but we don’t know yet whether that will happen.
We have some specific ideas of things we’d like to try, such as hybrid quantum-classicalalgorithms for solving optimization problems, both classical and quantum ones.
We can expect that, once we have quantum computers to experiment with, the development of quantum algorithms will accelerate. Perhaps new heuristics will be discovered for solving useful problems, although we may not be able to explain why theseheuristics work so well, at least not right away.
Although we probably won’t be able to use full-blown quantum error correction to protect
NISQ-era devices from noise, we should design near-term quantum algorithms withnoise resilience in mind. Clever design of noise-resilient algorithms may extend thecomputational power of NISQ devices.
Classical computers are especially bad at simulating the dynamics of highly entangled
many-particle quantum systems; therefore quantum dynamics is a particularly promising arena where quantum computers may have a significant advantage over classicalones.
We should continue to focus on building quantum hardware with lower gate error rates.
Improved quantum gate accuracy will lower the overhead cost of quantum error
correction when it is eventually implemented. In the nearer term, more accurate
gates will allow us to execute larger quantum circuits, and so extend the power of
NISQ technology.
NISQ is not likely to change the world all by itself. Instead, the primary goal for nearterm quantum platforms should be to pave the way for bigger payoffs which will be
realized by more advanced quantum devices down the road.

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15

The truly transformative quantum technologies of the future are probably going to haveto be fault tolerant, and because of the very hefty overhead cost of quantum error
correction, the era of fault-tolerant quantum computing may still be rather distant.
No one really knows how long it will take to get there. As the quantum community
eagerly seizes the impending opportunity to experiment with NISQ devices, we must
not lose sight of the essential longer-term goal: hastening the onset of the faulttolerant era.
I’ve adopted a cautionary stance in these remarks, because I think we should face theonset of the NISQ era with well-grounded expectations. But please don’t misunderstandme — like many of my colleagues, I’m eagerly anticipating the discoveries that will ensue as
the NISQ era unfolds over the next few years. Quantum technology is rife with exhilarating
opportunities, and surely many rousing surprises lie ahead. But the challenges we face arestill formidable. All quantumists should appreciate that our field can fulfill its potential
only through sustained, inspired effort over decades. If we pay that price, the ultimate
rewards will more than vindicate our efforts.

Acknowledgments
This article is based on a Keynote Address delivered at Quantum Computing for Businesson 5 December 2017. I thank Matt Johnson for organizing this stimulating meeting andinviting me to participate. My remarks here have been influenced by discussions withmany colleagues, too many to list. But I’ve especially benefited from insights due to Scott
Aaronson, Sergio Boixo, Fernando Brandão, Elizabeth Crosson, Toby Cubitt, Eddie Farhi,
Steve Flammia, David Gosset, Daniel Gottesman, Stephen Jordan, Jordan Kerenidis, Isaac
Kim, Seth Lloyd, Shaun Maguire, Oskar Painter, David Poulin, Peter Shor, Brian Swingle,
Matthias Troyer, Umesh Vazirani, and Thomas Vidick. Some of this work was done while
I attended the 2017 program on Quantum Physics of Information at the Kavli Institute for
Theoretical Physics (KITP). I gratefully acknowledge support from ARO, DOE, IARPA,
NSF, and the Simons Foundation. The Institute for Quantum Information and Matter(IQIM) is an NSF Physics Frontiers Center.

References[1] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev. 41, 303-332 (1999),
https://doi.org/10.1137/S0036144598347011.
[2] A. P. Lund, M. J. Bremner, and T. C. Ralph, Quantum sampling problems,
BosonSampling, and quantum supremacy, npj Quantum Information 3: 15 (2017),
arXiv:1702.03061, https://doi.org/10.1038/s41534-017-0018-2.
[3] A. W. Harrow and A. Montanaro, Quantum computational supremacy, Nature 549,
203-209 (2017), https://doi.org/10.1038/nature23458.
[4] S. P. Jordan, Quantum algorithm zoo, http://math.nist.gov/quantum/zoo/.
[5] A. Montanaro, Quantum algorithms: an overview, npj Quantum Information, 15023(2016), arXiv:1511.04206, https://doi.org/10.1038/npjqi.2015.23.
[6] L. Grover, Quantum mechanics helps in searching for a needle in ahaystack,
Phys. Rev. Lett. 79,
325 (1997),
arXiv:quant-ph/9706033,
https://doi.org/10.1103/PhysRevLett.79.325.
Accepted in Quantum 2018-07-30, click title to verify16[7] C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput. 26, 1510-1523 (1997), arXiv:quantph/9701001, https://doi.org/10.1137/S0097539796300933.
[8] R. B. Laughlin and D. Pines, The theory of everything, PNAS 97, 28-31 (2000),
https://doi.org/10.1073/pnas.97.1.28.
[9] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Physics 21, 467-488(1982).
[10] D. Gottesman, An introduction to quantum error correction and fault-tolerant quantum computation, Proceedings of Symposia in Applied Matthematics 68 (2010),
arXiv:0904.2557.
[11] S. Boixo, S. V. Isakov, V. N. Smelyansky, R. Babbush, N. Ding, Z. Jiang, M. J.
Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy innear-term devices, Nature Physics 14, 595-600 (2018), arXiv:1608.00263 (2016),
https://doi.org/10.1038/s41567-018-0124-x.
[12] S. Aaronson and L. Chen, Complexity-theoretic foundations of quantum supremacy
experiments, arXiv:1612.05903 (2017).
[13] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, T. Magerlein, E. Solomonik,
and R. Wisnieff, Breaking the 49-qubit barrier in the simulation of quantum circuits,
arXiv:1710.05867 (2017).
[14] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, High-fidelity
quantum logic gates using trapped-ion hyperfine qubits, Phys. Rev. Lett. 117, 060504(2016), arXiv:1512.04600, https://doi.org/10.1103/PhysRevLett.117.060504.
[15] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J.
Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,
C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov,
A. N. Cleland, and J. M. Martinis, Superconducting quantum circuits at the surface code threshold for fault tolerance, Nature 508, 500-503 (2014), arXiv:1402.4848,
https://doi.org/10.1038/nature13171.
[16] D. J. Bernstein, J. Buchmann, E. Dahmen, editors, Post-Quantum Cryptography,
Springer (2009), https:/doi.org/10.1007/978-3-540-88702-7.
[17] R. Alléaume, C. Branciard, J. Bouda, T. Debuisschert, M. Dianati, N. Gisin, M.
Godfrey, P. Grangier, T. Länger, N. Lütkenhaus, C. Monyk, P. Painchault, M. Peev, A.
Poppe, T. Pornin, J. Rarity, R. Renner, G. Ribordy, M. Riguidel, L. Salvail, A. Shields,
H. Weinfurter, and A. Zeilinger, Using quantum key distribution for cryptographic
purposes: a survey, Theoretical Computer Science 560, 62-81 (2014), arXiv:quantph/0701168, https://doi.org/10.1016/j.tcs.2014.09.018.
[18] S. Muralidharan, L. Li, J. Kim, N Lütkenhaus, M. D. Lukin, and L. Jiang, Efficient long distance quantum communication, Scientific Reports 6, 20463 (2016),
arXiv:1509.08435, https://doi.org/10.1038/srep20463.
[19] P. Bierhorst, E. Knill, S. Glancy, Y. Zhang, A. Mink, S. Jordan, A. Rommal, Y.-K.
Liu, B. Christensen, S. W. Nam, M. J. Stevens, and L. K. Shalm, Experimentally
generated randomness certified by the impossibility of superluminal signals, Nature
556, 223-226 (2018), arXiv:1803.06219, https://doi.org/10.1038/s41586-018-0019-0.
[20] Z. Brakerski, P. Christiano, U. Mahadev, U. Vazirani, and T. Vidick, Certifiablerandomness from a single quantum device, arXiv:1804.00640 (2018).
Accepted in Quantum 2018-07-30, click title to verify17[21] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89,
035002 (2017), arXiv:1611.04691, https://doi.org/10.1103/RevModPhys.89.035002.
[22] J. Preskill, Quantum computing and the entanglement frontier, 25th Solvay Conference on Physics (2011), arXiv:1203.5813.
[23] S. Khot, Hardness of approximation, Proceedings of the International Congress ofMathematicians (2014).
[24] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028 (2014).
[25] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory ofvariational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016),
arXiv:1509.04279, https://doi.org/10.1038/ncomms5213.
[26] D. A. Spielman and S.-H. Teng, Smoothed analysis of algorithms: why the simplex
algorithm usually takes polynomial time, Journal of the ACM 51, 385-463 (2004),
arXiv:cs/0111050, https://doi.org/10.1145/990308.990310.
[27] Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, Nature 521, 436-444 (2015),
https://doi.org/10.1038/nature14539.
[28] T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis,
D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345,
420-424 (2014), https://doi.org/10.1126/science.1252319.
[29] S. Mandrà, H. G. Katzgraber, and C. Thomas, The pitfalls of planar spin-glass benchmarks: raising the bar for quantum annealers (again), Quantum Sci. Technol. 2,038501 (2017), arXiv:1703.00622, https://doi.org/10.1088/2058-9565/aa7877.
[30] T. Albash and D. A. Lidar, Adiabatic quantum computing, Rev. Mod. Phys. 90,
015002 (2018), arXiv:1611.04471, https://doi.org/10.1103/RevModPhys.90.015002.
[31] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic
quantum computation is equivalent to standard quantum computation, SIAM Rev.
50, 755-787 (2008), arXiv:quant-ph/0405098.
[32] S. Bravyi, D. DiVincenzo, R. I. Oliveira, and B. M. Terhal, The complexity of stoquastic local Hamiltonian problems, Quant. Inf. Comp. 8, 0361-0385 (2008), arXiv:quantph/0606140.
[33] M. Jarret, S. P. Jordan, and B. Lackey, Adiabatic optimization versus diffusion Monte Carlo, Phys. Rev. A 94, 042318 (2016), arXiv:1607.03389,
https://doi.org/10.1103/PhysRevA.94.042318.
[34] A. D. King, J. Carrasquilla, I. Ozfidan, J. Raymond, E. Andriyash, A. Berkley, M.
Reis, T. M. Lanting, R. Harris, G. Poulin-Lamarre, A. Yu. Smirnov, C. Rich, F.
Altomare, P. Bunyk, J. Whittaker, L. Swenson, E. Hoskinson, Y. Sato, M. Volkmann,
E. Ladizinsky, M. Johnson, J. Hilton, and M. H. Amin, Observation of topologicalphenomena in a programmable lattice of 1,800 qubits, arXiv:1803.02047 (2018).
[35] I. H. Kim, Noise-resilient preparation of quantum many-body ground states,
arXiv:1703.00032 (2017).
[36] I. H. Kim and B. Swingle, Robust entanglement renormalization on a noisy quantum
computer, arXiv:1711.07500 (2017).
[37] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S.
Lloyd, Quantum machine learning, Nature 549, 195-202 (2017), arXiv:1611.09347,
https://doi.org/10.1038/nature23474.
Accepted in Quantum 2018-07-30, click title to verify

18[38] S. Aaronson, Read the fine print,
https://doi.org/10.1038/nphys3272.

Nature Physics 11,

291-293 (2015),[39] X. Gao, Z. Zhang, and L. Duan, An efficient quantum algorithm for generative machinelearning, arXiv:1711.02038 (2017).
[40] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear
systems of equations, Phys. Rev. Lett. 103, 150502 (2009), arXiv:0811.3171,
https://doi.org/10.1103/PhysRevLett.103.150502.
[41] B. D. Clader, B. C. Jacobs, and C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett. 110, 250504 (2013), arXiv:1301.2340,
https://doi.org/10.1103/PhysRevLett.110.250504.
[42] A. Montanaro and S. Pallister, Quantum algorithms and the finite element method, Phys. Rev. A 93, 032324 (2016), arXiv:1512.05903,
https://doi.org/10.1103/PhysRevA.93.032324.
[43] P. C. S. Costa, S. Jordan, and A. Ostrander, Quantum algorithm for simulating thewave equation, arXiv:1711.05394 (2017).
[44] I. Kerenidis and A. Prakash, Quantum recommendation systems, arXiv:1603.08675(2016).
[45] E. Tang, A quantum-inspired classical algorithm for recommendation systems, Electronic Colloquium on Computational Complexity, TR18-12 (2018).
[46] F. G. S. L. Brandão and K. Svore, Quantum speed-ups for semidefinite programming,
Proceedings of FOCS 2017, arXiv:1609.05537 (2017).
[47] F. G. S. L. Brandão, A. Kalev, T. Li, C. Y.-Y. Lin, K. M. Svore, and X. Wu, Exponential quantum speed-ups for semidefinite programming with applications to quantum
learning, arXiv:1710,02581 (2017).
[48] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction
mechanisms on quantum computers, PNAS 117, 7555-7560 (2017), arXiv:1605.03590,
https://doi.org/10.1073/pnas.1619152114.
[49] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Solving
strongly correlated electron models on a quantum computer, Phys. Rev. A 92, 062310(2015), arXiv:1506.05135, https://doi.org/10.1103/PhysRevA.92.062318.
[50] J. Olson, Y. Cao, J. Romero, P. Johnson, P.-L. Dallaire-Demers, N. Sawaya, P. Narang,
I. Kivlichan, M. Wasielewski, A. Aspuru-Guzik, Quantum information and computation for chemistry, NSF Workshop Report, arXiv:1706.05413 (2017).
[51] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi,
A. S. Zibrov, M. Endres, M. Greiner, V Vuletić, and M. D. Lukin, Probing manybody dynamics on a 51-atom quantum simulator, Nature 551, 579-584 (2017),
arXiv:1707.04344, https://doi.org/10.1038/nature24622.
[52] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V.
Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, arXiv:1708.01044 (2017),
https://doi.org/10.1038/nature24654.
[53] E. T. Campbell, B. M. Terhal, and C. Vuillot, The steep road towards robust anduniversal quantum computation, arXiv:1612.07330 (2016).

Accepted in Quantum 2018-07-30, click title to verify

19[54] J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016), arXiv:1512:01098,
https://doi.org/10.1103/PhysRevA.94.052325.
[55] J. Combes, C. Granade, C. Ferrie, and S. T. Flammia, Logical randomized benchmarking, arXiv:1702.03688 (2017).
[56] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes:
towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012),
arXiv:1208.0928, https://doi.org/10.1103/PhysRevA.86.032324.
[57] S. Das Sarma, M. Freedman, and C. Nayak, Majorana zero modes and topologicalquantum computation, npj Quantum Information 1, 15001 (2015), arXiv:1501.02813,
https://doi.org/10.1038/npjqi.2015.1.

Accepted in Quantum 2018-07-30, click title to verify

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