The Quantum Computer PuzzleGil KalaiCommunicated by Joel Hass

Q uantum computers are hypothetical devices,based on quantum physics, which wouldenable us to perform certain computationshundreds of orders ofmagnitude faster thandigital computers. This feature is coined

“quantum supremacy”, and one aspect or another of suchquantum computational supremacy might be seen byexperiments in the near future: by implementing quantumerror-correction or by systems of noninteracting bosonsor by exotic new phases of matter called anyons orby quantum annealing, or in various other ways. Weconcentrate in this paper on the model of a universalquantum computer that allows the full computationalpotential for quantum systems, and on the restrictedmodel, called “BosonSampling”, based on noninteractingbosons.

A main reason for concern regarding the feasibilityof quantum computers is that quantum systems areinherently noisy. We will describe an optimistic hypoth-esis regarding quantum noise that will allow quantumcomputing and a pessimistic hypothesis that won’t. The

Gil Kalai is Henry and Manya Noskwith Professor of Mathematicsat the Hebrew University of Jerusalem and is also affiliated withYale University. His email address is gil.kalai@gmail.com.

This work was supported in part by ERC advanced grant 320924,BSF grant 2006066, and NSF grant DMS-1300120. The authoris thankful to an anonymous referee, Bill Casselman, Irit Dinur,Oded Goldreich, Joel Hass, and Abby Thompson for helpful com-ments, and to Neta Kalai for drawing Figures 2 and 4.1Based on G. Kalai and G. Kindler, “Gaussian noise sensitivity andBosonSampling”, arXiv:1409.3093.2Based on G. Kalai, “How quantum computers fail: quantumcodes, correlations in physical systems, and noise accumulation”,arXiv:1106.0485, and a subsequent Internet debate with AramHarrow and others.

For permission to reprint this article, please contact: reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1380

quantum computer puzzle is to decide between thesetwo hypotheses. We list some remarkable consequencesof the optimistic hypothesis, giving strong reasons forthe intensive efforts to build quantum computers, aswell as good reasons for suspecting that this mightnot be possible. For systems of noninteracting bosons,we explain how quantum supremacy achieved withoutnoise is replaced, in the presence of noise, by a verylow yet fascinating computational power.1 Finally, wedescribe eight predictions about quantum physics andcomputation from the pessimistic hypothesis.2

Are quantum computers feasible? Is quantumsupremacy possible? My expectation is that the pes-simistic hypothesis will prevail, leading to a negativeanswer. Rather than regarding this possibility as an un-fortunate failure that impedes the progress of humanity,I believe that the failure of quantum supremacy itselfleads to important consequences for quantum physics,the theory of computing, andmathematics. Some of thesewill be explored here.

A Brief SummaryHere is a brief summary of the author’s pessimisticpoint of view as explained in the paper: understandingquantum computers in the presence of noise requiresconsideration of behavior at different scales. In the smallscale, standard models of noise from the mid-90s aresuitable, and quantum evolutions and states describedby them manifest a very low-level computational power.This small-scale behavior has far-reaching consequencesfor the behavior of noisy quantum systems at largerscales. On the one hand, it does not allow reaching thestarting points for quantum fault tolerance and quantumsupremacy, making them both impossible at all scales.On the other hand, it leads to novel implicit ways formodeling noise at larger scales and to various predictionson the behavior of noisy quantum systems.

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The Vision of Quantum Computers and QuantumSupremacyCircuits and Quantum CircuitsThe basic memory component in classical computing isa bit, which can be in two states, “0” or “1”. A computer(or circuit) has 𝑛 bits, and it can perform certain logicaloperations on them. The NOT gate, acting on a single bit,and the AND gate, acting on two bits, suffice for universalclassical computing. Thismeans that a computation basedon another collection of logical gates, each acting on aboundednumber of bits, canbe replacedby a computationbased only on NOT and AND. Classical circuits equippedwith random bits lead to randomized algorithms, whichare both practically useful and theoretically important.

Quantum computers (or circuits) allow the creationof probability distributions that are well beyond thereach of classical computers with access to random bits.A qubit is a piece of quantum memory. The state ofa qubit can be described by a unit vector in a two-dimensional complex Hilbert space 𝐻. For example, abasis for 𝐻 can correspond to two energy levels of thehydrogen atom or to horizontal and vertical polarizationsof a photon. Quantum mechanics allows the qubit tobe in a superposition of the basis vectors, describedby an arbitrary unit vector in 𝐻. The memory of aquantum computer consists of 𝑛 qubits. Let 𝐻𝑘 be thetwo-dimensional Hilbert space associated with the 𝑘thqubit. The state of the entire memory of 𝑛 qubits isdescribed by a unit vector in the tensor product 𝐻1 ⊗𝐻2 ⊗ ⋯ ⊗ 𝐻𝑛. We can put one or two qubits throughgates representing unitary transformations acting on thecorresponding two- or four-dimensional Hilbert spaces,and as for classical computers, there is a small list of gatessufficient for universal quantum computing. Each step inthe computation process consists of applying a unitarytransformation on the large 2𝑛-dimensional Hilbert space,namely, applying a gate on one or two qubits, tensoredwith the identity transformation on all other qubits. Atthe end of the computation process, the state of theentire computer can be measured, giving a probabilitydistribution on 0–1 vectors of length 𝑛.

A few words on the connection between the mathe-matical model of quantum circuits and quantum physics:In quantum physics, states and their evolutions (the waythey change in time) are governed by the Schrödingerequation. A solution of the Schrödinger equation can bedescribed as a unitary process on a Hilbert space, andquantum computing processes as we just described forma large class of such quantum evolutions.

A Very Brief Tour of Computational ComplexityComputational complexity is the theory of efficient compu-tations, where “efficient” is an asymptotic notion referringto situations where the number of computation steps(“time”) is at most a polynomial in the number of inputbits. The complexity class P is the class of algorithms thatcan be performed using a polynomial number of steps inthe size of the input. The complexity class NP refers tonondeterministic polynomial time. Roughly speaking, it

Figure 1. The (conjectured) view of some maincomputational complexity classes. The red ellipserepresents efficient quantum algorithms.

refers to questions where we can provably perform thetask in a polynomial number of operations in the inputsize, provided we are given a certain polynomial-size“hint” of the solution. An algorithmic task 𝐴 is NP-hardif a subroutine for solving 𝐴 allows solving any problemin NP in a polynomial number of steps. An NP-completeproblem is an NP-hard problem in NP. A useful analog isto think about the gap between NP and P as similar to thegap between finding a proof of a theorem and verifyingthat a given proof of the theorem is correct. P and NP aretwo of the lowest computational complexity classes in thepolynomial hierarchy PH, which is a countable sequenceof such classes, and there is a rich theory of complexityclasses beyond PH.

There are intermediate problems between P and NP.Factoring an 𝑛-digit integer is not known to be in P, asthe best algorithms are exponential in the cube root ofthe number of digits. Factoring is in NP, but it is unlikelythat factoring is NP-complete. Shor’s famous algorithmshows that quantum computers can factor 𝑛-digit inte-gers efficiently—in ∼ 𝑛2 steps! Quantum computers arenot known to be able to solve efficiently NP-completeproblems, and there are good reasons to think that theycannot. Yet, quantum computers can efficiently performcertain computational tasks beyond NP.

Two comments: First, our understanding of the com-putational complexity world depends on a whole arrayof conjectures: NP ≠ P is the most famous one, and astronger conjecture asserts that PH does not collapse,namely, that there is a strict inclusion between the com-putational complexity classes defining the polynomialhierarchy. Second, computational complexity insights,while asymptotic, strongly apply to finite and small algo-rithmic tasks. Paul Erdős famously claimed that findingthe value of the Ramsey function 𝑅(𝑛,𝑛) for 𝑛 = 6 is wellbeyond mankind’s ability. This statement is supportedby computational complexity insights that consider thedifficulty of computations as 𝑛 → ∞, while not directlyimplied by them.

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NoiseNoise and Fault-Tolerant ComputationThe main concern regarding the feasibility of quantumcomputers has always been that quantum systems areinherently noisy: we cannot accurately control them, andwe cannot accurately describe them. To overcome thisdifficulty, a theory of quantum fault-tolerant computationbased on quantumerror-correction codeswas developed.3Fault-tolerant computation refers to computation in thepresence of errors. The basic idea is to represent (or“encode”) a single piece of information (a bit in theclassical case or a qubit in the quantum case) by a largenumber of physical components so as to ensure that thecomputation is robust even if some of these physicalcomponents are faulty.

The main concernregarding thefeasibility ofquantum

computers hasalways been thatquantum systemsare inherently

noisy.

What is noise?Solutions of theSchrödinger equation(quantum evolutions)can be regarded asunitary processes onHilbert spaces. Mathe-matically speaking, thestudy of noisy quan-tum systems is thestudy of pairs ofHilbert spaces (𝐻,𝐻′),𝐻 ⊂ 𝐻′, and a uni-tary process on thelarger Hilbert space𝐻′. Noise refers tothe general effect ofneglecting degrees of

freedom, namely, approximating the process on a largeHilbert space by a process on a small Hilbert space. Forcontrolled quantum systems and, in particular, quantumcomputers, 𝐻 represents the controlled part of the sys-tem, and the large unitary process on 𝐻′ represents, inaddition to an “intended” controlled evolution on 𝐻, alsothe uncontrolled effects of the environment. The study ofnoise is relevant not only to controlled quantum systemsbut also to many other aspects of quantum physics.

A second, mathematically equivalent way to view noisystates and noisy evolutions is to stay with the originalHilbert space 𝐻 but to consider a mathematically largerclass of states and operations. In this view, the stateof a noisy qubit is described as a classical probabilitydistribution on unit vectors of the associated Hilbertspaces. Such states are referred to as mixed states. It isconvenient to think about the following form of noise,called depolarizing noise: in every computer cycle a qubitis not affected with probability 1−𝑝, and, with probability𝑝, it turns into the maximal entropy mixed state, i.e., theaverage of all unit vectors in the associated Hilbert space.In this example, 𝑝 is the error rate, and, more generally,the error rate can be defined as the probability that a

3M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information, Cambridge University Press, Cambridge,2000, Ch. 10.

qubit is corrupted at a computation step conditioned onit surviving up to this step.

Two Alternatives for Noisy Quantum SystemsThe quantum computer puzzle is, in a nutshell, de-ciding between two hypotheses regarding properties ofnoisy quantum circuits: the optimistic hypothesis and thepessimistic hypothesis.

Optimistic Hypothesis: It is possible to realize universalquantum circuits with a small bounded error level re-gardless of the number of qubits. The effort requiredto obtain a bounded error level for universal quantumcircuits increases moderately with the number of qubits.Therefore, large-scale fault-tolerant quantum computersare possible.

Pessimistic Hypothesis: The error rate in every realizationof a universal quantum circuit scales up (at least) linearlywith the number of qubits. The effort required to obtain abounded error level for any implementation of universalquantum circuits increases (at least) exponentially withthe number of qubits. Thus, quantum computers are notpossible.

Some explanations: For the optimistic hypothesis, wenote that the main theorem of quantum fault toleranceasserts that (under some natural conditions on the noise)if we can realize universal quantum circuits with asufficiently small error rate (where the threshold is roughlybetween 0.001 and 0.01), then quantum fault toleranceand hence universal quantum computing are possible.For the pessimistic hypothesis, when we say that the rateof noise per qubit scales up linearly with the numberof qubits, we mean that when we double the number ofqubits in the circuit, the probability for a single qubitto be corrupted in a small time interval doubles. Thepessimistic hypothesis does not require new modelingfor the noise for universal quantum circuits, and it isjust based on a different assumption on the rate ofnoise. However, it leads to interesting predictions andmodeling and may lead to useful computational tools,for more general noisy quantum systems. We emphasizethat both hypotheses are assertions about physics (orphysical reality), not about mathematics, and both of thehypotheses represent scenarios that are compatible withquantum mechanics.

The constants are important, and the pessimistic viewregarding quantum supremacy holds that every realiza-tion of universal quantum circuits will fail for a handfulof qubits long before any quantum supremacy effect iswitnessed and long before quantum fault tolerance ispossible. The failure to reach universal quantum circuitsfor a small number of qubits and to manifest quantumsupremacy for small quantum systems is crucial forthe pessimistic hypothesis, and Erdős’s statement about𝑅(6, 6) is a good analogy for this expected behavior.

Both on the technical and conceptual levels we see herewhat we call a “wide-gap dichotomy”. On the technicallevel, we have a gap between small constant error rate perqubit for the optimistic view and linear increase of rate

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Figure 2. The optimistic hypothesis: Classicalfault-tolerance mechanisms can be extended, viaquantum error-correction, allowing robust quantuminformation and computationally superior quantumcomputation. Drawing by Neta Kalai.

per qubit (in terms of the number of qubits in the circuit)on the pessimistic side. We also have a gap betweenthe ability to achieve large-scale quantum computers onthe optimistic side and the failure of universal quantumcircuits already for a handful of qubits on the pessimisticside. On the conceptual level, the optimistic hypothesisasserts that quantum mechanics allows superior compu-tational powers, while the pessimistic hypothesis assertsthat quantum systems without specific mechanisms forrobust classical information that leads only to classicalcomputing are actually computationally inferior. We willcome back to both aspects of this wide-gap dichotomy.

Potential Experimental Support for QuantumSupremacyA definite demonstration of quantum supremacy ofcontrolled quantum systems—namely, building quantumsystems that outperform, even for specific computationaltasks, classical computers—or a definite demonstrationof quantum error correction will falsify the pessimistichypothesis and will give strong support for the optimistichypothesis. (The optimistic hypothesis will be completelyverified with full-fledged universal quantum computers.)There are several ways people plan, in the next few years,to demonstrate quantum supremacy or the feasibility ofquantum fault tolerance.

(1) Attempts to create small universal quantumcircuits with up to “a few tens of qubits.”

(2) Attempts to create stable logical qubits based onsurface codes.

(3) Attempts to have BosonSampling for 10–50bosons.

(4) Attempts to create stable qubits based on anyonicstates.

(5) Attempts to demonstrate quantum speed upbased on quantum annealing.

Each of attempts (1)–(4) represents many differentexperimental directions carried out mainly in academicinstitutions, while (5) represents an attempt by a commer-cial company, D-wave.4 There are many different avenuesfor realizing qubits, of which ion-trapped qubits andsuperconducting qubits are perhaps the leading ones.Quantum supremacy via nonabelian anyons stands out asa very different direction based on exotic new phases ofmatter and very deep mathematical and physical issues.BosonSampling (see the next section) stands out in thequest to demonstrate quantum supremacy for narrowphysical systems without offering further practical fruits.

The pessimistic hypothesis predicts a decisive fail-ure for all of these attempts to demonstrate quantumsupremacy or very stable logical qubits and that thisfailure will be witnessed for small systems. A readermay ask how the optimistic hypothesis can be falsifiedbeyond repeated failures to demonstrate universal quan-tum computers or partial steps toward them as thoselisted above. My view is that the optimistic hypothesiscan be largely falsified if we can understand the absenceof quantum supremacy and quantum error correctionas a physical principle with predictive power that goesbeyond these repeated failures, both in providing moredetailed predictions about these failures themselves (suchas scaling-up of errors, correlations between errors, etc.)and in providing predictions for other natural quantumsystems. Mathematical modeling of noisy quantum sys-tems based on the pessimistic hypothesis is valuable, notonly if it represents a general physical principle, but alsoif it represents temporary technological difficulties or ifit applies to limited classes of quantum systems.

BosonSamplingQuantum computers allow the creation of probabilitydistributions that are beyond the reach of classical com-puters with access to random bits. This is manifestedby BosonSampling, a class of probability distributionsrepresenting a collection of noninteracting bosons thatquantum computers can efficiently create. It is a restrictedsubset of distributions compared to the class of distri-butions that a universal quantum computer can produce,and it is not known if BosonSampling distributions can beused for efficient integer factoring or for other “useful”algorithms. BosonSampling was introduced by Troyan-sky and Tishby in 1996 and was intensively studied byAaronson and Arkhipov,5 who offered it as a quick pathfor experimentally showing that quantum supremacy is areal phenomenon.

Given an 𝑛 by 𝑛 matrix 𝐴, let 𝑑𝑒𝑡(𝐴) denote thedeterminant of 𝐴, and let 𝑝𝑒𝑟(𝐴) denote the perma-nent of 𝐴. Thus 𝑑𝑒𝑡(𝐴) = ∑𝜋∈𝑆𝑛 𝑠𝑔𝑛(𝜋)∏𝑛

𝑖=1 𝑎𝑖𝜋(𝑖), and𝑝𝑒𝑟(𝐴) = ∑𝜋∈𝑆𝑛 ∏𝑛

𝑖=1 𝑎𝑖𝜋(𝑖). Let 𝑀 be a complex 𝑛 × 𝑚

4D-wave is attempting to demonstrate quantum speedup forNP-hard optimization problems and even to compute Ramseynumbers.5S. Aaronson and A. Arkhipov, “The computational complex-ity of linear optics”, Theory of Computing 4 (2013), 143–252;arXiv:1011.3245.

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matrix, 𝑚 ≥ 𝑛. Consider all (𝑚𝑛) subsets 𝑆 of 𝑛 columns,and for every subset consider the corresponding 𝑛 × 𝑛submatrix 𝐴. The algorithmic task of sampling subsets𝑆 of columns according to |𝑑𝑒𝑡(𝑀′)|2 is called Fermion-Sampling. Next consider all (𝑚+𝑛−1

𝑛 ) submultisets 𝑆 of 𝑛columns (namely, allow columns to repeat), and for everysubmultiset𝑆 consider the corresponding𝑛×𝑛 submatrix𝐴 (with column 𝑖 repeating 𝑟𝑖 times). BosonSampling is thealgorithmic task of sampling those multisets 𝑆 accordingto |𝑝𝑒𝑟(𝐴)|2/(𝑟1! 𝑟2!⋯𝑟𝑛! ). Note that the algorithmic taskfor BosonSampling and FermionSampling is to sampleaccording to a specified probability distribution. They arenot decision problems, where the algorithmic task is toprovide a yes/no answer.

Let us demonstrate these notions by an example for𝑛 = 2 and 𝑚 = 3. The input is a 2 × 3 matrix:

( 1/√3 𝑖/√3 1/√30 1/√2 𝑖/√2 ) .

The output for FermionSampling is a probability dis-tribution on subsets of two columns, with probabilitiesgiven according to absolute values of the square of deter-minants. Here we have probability 1/6 for columns {1, 2},probability 1/6 for columns {1, 3}, and probability 4/6 forcolumns {2, 3}. The output for BosonSampling is a prob-ability distribution according to absolute values of thesquare of permanents of submultisets of two columns.Here, the probabilities are: {1, 1} → 0, {1, 2} → 1/6,{1, 3} → 1/6, {2, 2} → 2/6, {2, 3} → 0, {3, 3} → 2/6.

FermionSampling describes the state of 𝑛 noninteract-ing fermions, where each individual fermion is describedas a superposition of 𝑚 “modes”. BosonSampling de-scribes the state of 𝑛 noninteracting fermions, whereeach individual fermion is described by 𝑚 modes. Afew words about the physics: Fermions and bosons arethe main building blocks of nature. Fermions, such aselectrons, quarks, protons, and neutrons, are particlescharacterized by Fermi–Dirac statistics. Bosons, suchas photons, gluons, and the Higgs boson, are particlescharacterized by Bose–Einstein statistics.

Moving to computational complexity, we note thatGaussian elimination gives an efficient algorithm forcomputing determinants, but computing permanents isvery hard: it represents a computational complexity classcalled #P (in words, “number P” or “sharp P”) that extendsbeyond the entire polynomial hierarchy. It is commonlybelieved that even quantum computers cannot efficientlycompute permanents. However, a quantum computer canefficiently create abosonic (anda fermionic) state basedona matrix 𝑀 and therefore perform efficiently both Boson-Sampling and FermionSampling. A classical computerwith access to random bits can sample FermionSamplingefficiently, but, as proved by Aaronson and Arkhipov, aclassical computer with access to random bits cannotperform BosonSampling unless the polynomial hierarchycollapses!

Predictions from the Optimistic HypothesisBarriers Crossed. Quantum computers would dramati-cally change our reality.

(1) A universal machine for creating quantum statesand evolutions will be built.

(2) Complicated evolutions and states with globalinteractions, markedly different from anythingwitnessed so far, will be created.

(3) It will be possible to experimentally time-reverseevery quantum evolution.

(4) The noise will not respect symmetries of the state.(5) There will be fantastic computational complexity

consequences.(6) Quantum computers will efficiently break most

current public-key cryptosystems.Items (1)–(4) represent a vastly different experimental

reality than that of today, and items (5) and (6) representa vastly different computational reality.

Magnitude of Improvements. It is often claimed that quan-tum computers can perform certain computations thateven a classical computer of the size of the entire uni-verse cannot perform! Indeed it is useful to examinenot only things that were previously impossible and thatare now made possible by a new technology but alsothe improvement in terms of orders of magnitude fortasks that could have been achieved by the old technology.Quantum computers represent enormous, unprecedentedorder-of-magnitude improvement of controlled physicalphenomena as well as of algorithms. Nuclear weaponsrepresent an improvement of 6–7 orders of magnitudeover conventional ordnance: the first atomic bomb wasa million times stronger than the most powerful (single)conventional bomb at the time. The telegraph could de-liver a transatlanticmessage in a few seconds compared tothe previous three-month period. This represents an (im-mense) improvement of 4–5 orders ofmagnitude. Memoryand speed of computers were improved by 10–12 ordersof magnitude over several decades. Breakthrough algo-rithms at the time of their discovery also representedpractical improvements of no more than a few orders

Photoco

urtesy

ofYifanWan

g.

Aram Harrow and Gil Kalai shake hands at MIT aftertheir internet debate.

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of magnitude. Yet implementing BosonSampling with ahundred bosons represents more than a hundred ordersof magnitude improvement compared to digital comput-ers, and a similar story can be told about a large-scalequantum computer applying Shor’s algorithm.

Computations in Quantum Field Theory. Quantum elec-trodynamics (QED) computations allow one to describevarious physical quantities in terms of a power series

∑𝑐𝑘𝛼𝑘,where 𝑐𝑘 is the contribution of Feynman’s diagrams with 𝑘loops and 𝛼 is the fine structure constant (around 1/137).Quantum computers will (likely)6 allow one to computethese terms and sums for large values of 𝑘 with hundredsof digits of accuracy, similar to computations of the digitsof 𝑒 and 𝜋 on today’s computers, even in regimes wherethey have no physical meaning!

My Interpretation. I regard the incredible consequencesfrom the optimistic hypothesis as solid indications thatquantum supremacy is “too good to be true” and that thepessimistic hypothesis will prevail. Quantum computerswould change reality in unprecedented ways, both qualita-tively and quantitatively, and it is easier to believe that wewill witness substantial theoretical changes in modelingquantum noise than that we will witness such dramaticchanges in reality itself.

BosonSampling Meets RealityHow Does Noisy BosonSampling Behave?BosonSampling and Noisy BosonSampling (i.e., BosonSam-pling in the presence of noise) exhibit radically differentbehavior. BosonSampling is based on 𝑛 noninteracting,indistinguishable bosons with 𝑚 modes. For noisy BosonSamplers these bosons will not be perfectly noninter-acting (accounting for one form of noise) and will notbe perfectly indistinguishable (accounting for anotherform of noise). The same is true if we replace bosonsby fermions everywhere. The state of 𝑛 bosons with 𝑚modes is represented by an algebraic variety of decom-posable symmetric tensors of real dimension 2𝑚𝑛 in ahuge relevant Hilbert space of dimension 2𝑚𝑛. For thefermion case this manifold is simply the Grassmannian.

We have already discussed the rich theory of compu-tational complexity classes beyond P, and there is also arich theory below P. One very low-level complexity classconsists of computational tasks that can be carried outby bounded-depth polynomial-size circuits. In this modelthe number of gates is, as before, at most polynomialin the input side, but an additional severe restriction isthat the entire computation is carried out in a boundednumber of rounds. Bounded-depth polynomial-size cir-cuits cannot even compute or approximate the parity of𝑛 bits, but they can approximate real functions described

6This plausible conjecture, which motivated quantum computersto start with, is supported by the recent work of Jordan, Lee, andPreskill and is often taken for granted. A mathematical proof isstill beyond reach.

Figure 3. The huge computational gap (left) betweenBosonSampling (purple) and FermionSampling(green) vanishes in the noisy versions (right).

by bounded-degree polynomials and can sample approx-imately according to probability distributions describedby real polynomials of bounded degree.

Theorem 1 (Kalai and Kindler). When the noise level isconstant, BosonSampling distributions are well approx-imated by their low-degree Fourier–Hermite expansion.Consequently, noisy BosonSampling can be approximatedby bounded-depth polynomial-size circuits.

It is reasonable to assume that for all proposed im-plementations of BosonSampling, the noise level is atleast a constant, and therefore an experimental re-alization of BosonSampling represents, asymptotically,bounded-depth computation. The next theorem showsthat implementation of BosonSampling will actuallyrequire pushing down the noise level below 1/𝑛.Theorem 2 (Kalai and Kindler). When the noise level is𝜔(1/𝑛) and 𝑚 ≫ 𝑛2, BosonSampling is very sensitive tonoise, with a vanishing correlation between the noisy dis-tribution and the ideal distribution.7

Theorems 1 and 2 give evidence against expectations ofdemonstrating “quantum supremacy” via BosonSampling:experimental BosonSampling represents an extremelylow-level computation, and there is no precedence fora “bounded-depth machine” or a “bounded-depth algo-rithm” that gives a practical advantage, even for smallinput size, over the full power of classical computers, notto mention some superior powers.

Bounded-Degree PolynomialsThe class of probability distributions that can be ap-proximated by low-degree polynomials represents asevere restrictionbelowbounded-depth computation. Thedescription of noisy BosonSampling with low bounded-degree polynomials is likely to extend to small noisyquantum circuits and other similar quantum systems,and this would support the pessimistic hypothesis. This

7The condition 𝑚 ≫ 𝑛2 can probably be removed by a moredetailed analysis.

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description is relevant to important general computa-tional aspects of quantum systems in nature, as we nowdiscuss.

Why Is Robust Classical Information Possible? The abilityto approximate low-degree polynomials still supportsrobust classical information. The (“Majority”) Booleanfunction8𝑓(𝑥1, 𝑥2,… , 𝑥𝑛) = 𝑠𝑔𝑛(𝑥1+𝑥2+⋯+𝑥𝑛)allows forvery robust bits based on a large number of noisy bits andadmits excellent low-degree approximations. Quantumerror correction is also based on encoding a single qubitas a function 𝑓(𝑞1, 𝑞2 …,𝑞𝑛) of many qubits, and alsofor quantum codes the quality of the encoded qubitgrows with the number of qubits used for the encoding.But for quantum error-correction codes, implementationwith bounded-degree polynomial approximations is notavailable, and I conjecture that no such implementationexists. This would support the insight that quantummechanics is limiting the information one can extractfrom a physical system in the absence of mechanismsleading to robust classical information.

Why Can We Learn the Laws of Physics from Experiments?Learning the parameters of a process from examplescan be computationally intractable, even if the processbelongs to a low-level computational task. (Learning evena function described by a depth-two Boolean circuit ofpolynomial size does not admit an efficient algorithm.)However, the approximate value of a low-degree polyno-mial can efficiently be learned from examples. This offersan explanation for our ability to understand naturalprocesses and the parameters defining them.

Predictions from the Pessimistic HypothesisUnder the pessimistic hypothesis, universal quantumdevices are unavailable, and we need to devise a specificdevice in order to implement a specific quantumevolution.A sufficiently detailed modeling of the device will lead toa familiar detailed Hamiltonian modeling of the quantumprocess that also takes into account the environment andvarious forms of noise. Our goal is different: we wantto draw from the pessimistic hypothesis predictions onnoisy quantum circuits (and, at a later stage, on moregeneral noisy quantum processes) that are common to alldevices implementing the circuit (process).

The basic premises for studying noisy quantum evolu-tions when the specific quantum devices are not specifiedare as follows: First, modeling is implicit; namely, it isgiven in terms of conditions that the noisy process mustsatisfy. Second, there are systematic relations betweenthe noise and the entire quantum evolution and alsobetween the target state and the noise.

In this section we assume the pessimistic hypothesis,but we note that the previous section proposes thefollowing picture in support of the pessimistic hypothesis:evolutionsandstatesofquantumdevices in thesmall scaleare described by low-degree polynomials. This allows, fora larger scale, the creation of robust classical informationand computation but does not provide the necessary

8A Boolean function is a function from {−1, 1}𝑛 to {−1, 1}.

Figure 4. The pessimistic hypothesis: Noisy quantumevolutions, described by low-degree polynomials,allow via the mechanisms of averaging/repetitionrobust classical information and computation but donot allow reaching the starting points for quantumsupremacy and quantum fault tolerance. Drawing byNeta Kalai.

starting point for quantum fault tolerance or for anymanifestation of quantum supremacy.

NoQuantum Fault Tolerance: Its Simplest ManifestationEntanglement and Cat States. Entanglement is a name forquantum correlation, and it is an important feature ofquantum physics and a crucial ingredient of quantumcomputation. A cat state of the form 1

√2|00⟩ + 1

√2|11⟩

represents the simplest form of entanglement betweentwo qubits. Let me elaborate: the Hilbert space 𝐻 repre-senting the states of a single qubit is two-dimensional.We denote by |0⟩ and |1⟩ the two vectors of a basis for 𝐻.A pure state of a qubit is a superposition of basis vectorsof the form 𝑎 |0⟩ + 𝑏 |1⟩, where 𝑎,𝑏 are complex and|𝑎|2 + |𝑏|2 = 1. Two qubits are represented by a tensorproduct 𝐻⊗𝐻, and we denote it by |00⟩ = |0⟩⊗|0⟩. Now,a superposition of two vectors can be thought of as aquantum analog of a coin toss in classical probability—asuperposition of |00⟩ and |11⟩ is a quantum analog ofcorrelated coin tosses: two heads with probability 1/2,and two tails with probability 1/2. The name “cat state”refers, of course, to Schrödinger’s cat.

Noisy Cats. The following prediction regarding noisy en-tangled pairs of qubits (or “noisy cats”) is perhaps thesimplest prediction on noisy quantum circuits under thepessimistic hypothesis.Prediction 1: Two-qubits behavior. Any implementationof quantum circuits is subject to noise, for which errorsfor a pair of entangled qubitswill have substantial positivecorrelation.

Prediction 1, which we will refer to as the “noisy catprediction”, gives a very basic difference between the op-timistic and pessimistic hypotheses. Under the optimistichypothesis gated qubits will manifest correlated noise,

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but when quantum fault tolerance is in place, such corre-lations will be diminished for most pairs of qubits. Underthe pessimistic hypothesis quantum fault-tolerance is notpossible, and without it there is no mechanism to removecorrelated noise for entangled qubits. Note that the condi-tion on noise for a pair of entangled qubits is implicit, asit depends on the unknown process and unknown deviceleading to the entanglement.

Further Simple Manifestations of the Failure of QuantumFault Tolerance.Prediction 2: Error synchronization. For complicated(very entangled) target states, highly synchronized errorswill occur.

Error synchronization refers to a substantial proba-bility that a large number of qubits, much beyond theaverage rate of noise, are corrupted. Under the optimistichypothesis error synchronization is an extremely rareevent.Prediction 3: Error rate. For complicated evolutions, andfor evolutions approximating complicated states, theerror rate, in terms of qubit-errors, scales up linearly withthe number of qubits.

The three predictions 1–3 are related. Under naturalassumptions, the noisy cat prediction implies error syn-chronization for quantum states of the kind involvedin quantum error correction and quantum algorithms.Roughly speaking, the noisy cat prediction implies posi-tive correlation between errors for every pair of qubits,and this implies a substantial probability for the eventthat a large fraction of qubits (well above the average rateof errors) will be corrupted at the same computer cycle.Error synchronization also implies, again under somenatural assumptions, that error rate in terms of qubiterrors is at least linear in the number of qubits. Thus,the pessimistic hypothesis itself can be justified from thenoisy cat prediction, together with natural assumptionson the rate of noise. Moreover, this also explains thewide-gap dichotomy in terms of qubit errors.

The optimistic hypothesis allows creating via quantumerror correction very stable “logical” qubits based onstable raw physical qubits.Prediction 4: No logical qubits. Logical qubits cannot besubstantially more stable than the raw qubits used toconstruct them.

No Quantum Fault-Tolerance: Its Most GeneralManifestation9

We can go to the other extreme and try to examineconsequences of the pessimistic hypothesis for the mostgeneral quantum evolutions. We start with a predictionrelated to the discussion in the section “BosonSamplingMeets Reality”.Prediction 5: Bounded-depth and bounded-degree ap-proximations. Quantum states achievable by anyimplementation of quantum circuits are limited bybounded-depth polynomial-size quantum computation.

9This section is more technical and assumes more background onquantum information.

Even stronger: low-entropy quantum states in natureadmit approximations by bounded-degree polynomials.

The next items go beyond the quantum circuit modeland do not assume that the Hilbert space for our quantumevolution has a tensor product structure.Prediction 6: Time smoothing. Quantum evolutions aresubject to noise, with a substantial correlation withtime-smoothed evolutions.

Time-smoothed evolutions form an interesting re-stricted class of noisy quantum evolutions aimed tomodel evolutions under the pessimistic hypothesis whenfault tolerance is unavailable to suppress noise prop-agation. The basic example for time-smoothing is thefollowing: Start with an ideal quantum evolution given bya sequence of 𝑇 unitary operators, where 𝑈𝑡 denotes theunitary operator for the 𝑡th step, 𝑡 = 1, 2,… ,𝑇. For 𝑠 < 𝑡we denote 𝑈𝑠,𝑡 = ∏𝑡−1

𝑖=𝑠 𝑈𝑖 and let 𝑈𝑠,𝑠 = 𝐼 and 𝑈𝑡,𝑠 = 𝑈−1𝑠,𝑡 .

The next step is to add noise in a completely standardway: consider a noise operation 𝐸𝑡 for the 𝑡th step. Wecan think about the case where the unitary evolutionis a quantum computing process and 𝐸𝑡 represents adepolarizing noise with a fixed rate acting independentlyon the qubits. And finally, replace 𝐸𝑡 with a new noiseoperation 𝐸′

𝑡 defined as the average

(1) 𝐸′𝑡 =

1𝑇 ⋅

𝑇

∑𝑠=1

𝑈𝑠,𝑡𝐸𝑠𝑈−1𝑠,𝑡 .

Prediction 7: Rate. For a noisy quantum system a lowerbound for the rate of noise in a time interval is a measureof noncommutativity for the projections in the algebra ofunitary operators in that interval.

Predictions 6 and 7 are implicit and describe systematicrelations between the noise and the evolution. We expectthat time-smoothing will suppress high terms for someFourier-like expansion, thus relating Predictions 5 and 6.We also note that Prediction 7 resembles the picture aboutthe “unsharpness principle” from symplectic geometryand quantization.10

Locality, Space and TimeThe decision between the optimistic and pessimistic hy-potheses is, to a large extent, a question about modelinglocality in quantum physics. Modeling natural quantumevolutions by quantum computers represents the impor-tant physical principle of “locality”: quantum interactionsare limited to a few particles. The quantum circuit modelenforces local rules on quantum evolutions and stillallows the creation of very nonlocal quantum states.This remains true for noisy quantum circuits under theoptimistic hypothesis. The pessimistic hypothesis sug-gests that quantum supremacy is an artifact of incorrectmodeling of locality. We expect modeling based on thepessimistic hypothesis, which relates the laws of the“noise” to the laws of the “signal”, to imply a strong formof locality for both.

We can even propose that spacetime itself emergesfrom the absence of quantum fault tolerance. It is a

10L. Polterovich, “Symplectic geometry of quantum noise”, Comm.Math. Phys 327 (2014), 481–519; arXiv:1206.3707.

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familiar idea that since (noiseless) quantum systemsare time reversible, time emerges from quantum noise(decoherence). However, also in the presence of noise,with quantum fault tolerance, every quantum evolutionthat can experimentally be created can be time-reversed,and, in fact, we can time-permute the sequence of unitaryoperators describing the evolution in an arbitrary way.It is therefore both quantum noise and the absence ofquantum fault tolerance that enable an arrow of time.

Next, we note that with quantum computers one canemulate a quantum evolution on an arbitrary geometry.For example, a complicated quantum evolution represent-ing the dynamics of a four-dimensional lattice modelcould be emulated on a one-dimensional chain of qubits.This would be vastly different from today’s experimentalquantum physics, and it is also in tension with insightsfrom physics, where witnessing different geometries sup-porting the same physics is rare and important. Since auniversal quantum computer allows the breaking of theconnection between physics and geometry, it is noise andthe absence of quantum fault tolerance that distinguishphysical processes based on different geometries andenable geometry to emerge from the physics.

Classical Simulations of Quantum SystemsPrediction 8: Classical simulations of quantum pro-cesses. Computations in quantum physics can, inprinciple, be simulated efficiently on a digital computer.

This bold prediction from the pessimistic hypothesiscould lead to specific models and computational tools.There are some caveats: heavy computations may berequired for quantum processes that are not realistic tostart with, for a model in quantum physics representing aphysical process that depends on many more parametersthan those represented by the input size, for simulatingprocesses that require knowing internal parameters ofthe process that are not available to us (but are availableto nature), and when we simply do not know the correctmodel or relevant computational tool.

Photoco

urtesy

ofMatas

Šileikis.

Gil Kalai lecturing at Adam Mickiewicz University inPoland.

Quid est Noster Computationis Mundus?11Deciding between the optimistic and pessimistic hypothe-ses reflects a far-reaching difference in the view of ourcomputational world. Is the wealth of computations wewitness in reality only the tip of the iceberg of a supremecomputational power used by nature and available to us,or is it the case that the wealth of classical computa-tions we witness represents the full computational powerthat can be extracted from natural quantum physicsprocesses?

I expect that the pessimistic hypothesis will prevail,yielding important outcomes for physics, the theory ofcomputing, andmathematics. Our journey through proba-bility distributions described by low-degree polynomials,implicit modeling for noise, and error synchronizationmay provide some of the ingredients needed for solvingthe quantum computer puzzle.

11What is our computational world?

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