Noise Stability, Noise Sensitivity and the Quantum Computer Puzzle

A tour through models, interpretations, analogies and laws

Gil KalaiEinstein Institute of MathematicsHebrew University of Jerusalem

ICM 2018, beautiful Rio

Part I: Noise sensitivity and stability

Model 1: Boolean functions (2-candidate voting rule)

Boolean functions are functions f(x1,x2,…,xn) of n Boolean variables (xi=1 or xi=-1) so that the value of f is -1 or 1.

Boolean functions are of importance in combinatorics, probability theory, computer science, voting, and other areas.

Examples. Majority: n is odd, f=1 if x1+x2+…+xn > 0.Dictatorship: f=x1 .

For simplicity we consider only odd Boolean functions , namely those that satisfy f(-x)=-f(x).

What is a quasi-democracy?

Consider a society whose constitutional voting method is given by a sequence (fn ) of Boolean functions, where fn is a Boolean function with n variables. The society is quasi-democratic if, as n goes to infinity, the probability* of every voter to determine the outcome of the election when the other voters vote at random tends to 0.

* This probability is called the influence or the Banzhaf power index.

Majority is stablest for quasi-democracies

Theorem 1: (Mossel, O’Donnell and Oleszkiewicz 2005)Suppose that every voter votes at random for each of two candidates with probability ½ (independently). Next suppose that there is a probability t for a mistake in counting each vote. For every quasi-democratic society, the probability that the outcome of the election is reversed is at least

(1-o(1)) arccos(1-2t)/π.

For the majority rule we have equality by a 1899 result by Sheppard. acrcos (1-2t) behaves like t1/2 . The majority rule is noise stable, and no

quasi democratic rule is more noise stable.

Model 2: 3-candidate elections

Every voter has an order relation between three candidates. We use a 2-candidate voting rule to decide society’s preference relation.

Codorcet’s “Paradox”: The majority rule may lead to cyclic social preferences.

When there are many voters and voters’ preferences are random the probability for the paradox tends to 0.08874… (Guilbaud (1952)).

0.87856… and 0.08874…consequences of theorem 1

Corollary 1: 0.08874… is the minimum probability (as the number of voters tends to infinity) for cyclic outcomes for 3-candidate quasi democratic voting rules. Mossel, O’Donnell and Oleszkiewicz (2005). Attained by the majority rule (Guilbaud (1952)).

Corollary 2: 0.87856…is the best ratio for efficient algorithm for MAX-CUT (under unique game conjecture). Attained by Goemans-Williamson algorithm (1995).Khot, Kindler, O’Donnell, Mossel (2004) and Mossel, O’Donnell and Oleszkiewicz (2005)

Model 3: Critical planar percolation

Theorem 2 (Benjamini, Kalai, Schramm 1999): The crossing event for n by n board for planar critical percolation is noise-sensitive. Namely, for evert t>0, starting with a random coloring, if you switch the color of each hexagon with probability t, the effect is like random recoloring!

Much stronger versions were achieved by Schramm-Steif(2010) and Garban-Pete-Schramm (2010).

Noise-sensitivity of Percolation: the proof

I. Fourier

II. Noise sensitivity via Fourier

Noise-sensitivity of Percolation: the proof

Theorem 3(Benjamini, Kalai, Schramm 99): A sequence of monotone Boolean functions is noise sensitive unless it has a uniformly positive correlation with weighted majority rule.

III. Deeper use of Fourier methods

IV. Basic results on planar critical percolation.

The proof is completed by using results about planar percolation (Russo-Seymour-Welsh, Kesten).

Noise sensitivity and other models in mathematical

physics

Glazman, Harel, Journel, and Peled study noise sensitivity of the Planar Gaussian free field using both 2D Fourier transform and high dimensional Fourier-Hermiteexpansion. (Benjamini posed the question.)

Dynamic percolation. Haggstrom, Peres and Steif (1995); Benjamini (1992); Schramm and Steif (2010); Garban, Pete, and Schramm (2010,2018): The Hausdorf dimension of the exceptional times for critical dynamaical percolation is 31/36

First passage percolation

Planar Gaussian free field

Benjamini, Kalai, Schramm (2002), Benami Rossignol (2008), Chatterjee(2008) …

Source:Sheffield

Source: Garban, Pete Schramm

Model 4: Computers! (Boolean circuits)The basic memory component in classical computing is a bit, which can be in two states, “0” or “1”.

A computer (or circuit) has ! bits, and it can perform certain logical operations on them. The NOT gate, acting on a single bit, and the AND gate, acting on two bits, suffice for the full power of classical computing.

Classical circuits equipped with random bits lead to randomized algorithms, which are both practically useful and theoretically important

Part II: Computation

Efficient computation and Computational complexity

• The complexity class P refers to problems that can be solved using a polynomial number of steps in the size of the input.

• The complexity class NP refers to problems whose solution can be verified in polynomial number of steps.

• Our understanding of the computational complexity world depends on a whole array of conjectures: NP ≠ P is the most famous one.

• Shor’s famous algorithm shows that quantum computers can factor !-digit integers efficiently—in ∼ !2 steps!

Model 5: Quantum computers

Qubits are unit vectors in C2: A qubit is a piece of quantum

memory. The state of a qubit is a unit vector in a 2-dimensional

complex Hilbert space H = . The memory of a quantum computer

(quantum circuit) consists of n qubits and the state of the computer is

a unit vector in .

Gates are unitary transformations: We can put one or two qubits

through gates representing unitary transformations acting on the

corresponding two- or four-dimensional Hilbert spaces, and as for

classical computers, there is a small list of gates sufficient for the full

power quantum computing.

Measurement: Measuring the state of k qubits leads to a probability

distribution on 0–1 vectors of length k.

Efficient computation and Computational complexity

Model 6: Noisy quantum computers

A quantum circuit such that every qubit is corrupted in every “computer cycle” with a small probability t, and every gate is t-imperfect.

t is a small constant called the rate of noise.

Quantum systems are inherently noisy; we cannot accurately control them, and we cannot accurately describe them. In fact, every interaction of a quantum system with the outside world accounts for noise.

The Threshold Theorem

Theorem 4: If the error rate is small enough, noisy quantum circuits allows universal quantum computing.Aharonov, Ben-Or (1995), Kitaev (1995), Knill, Lafflamme, Zurek (1995), following Shor (1995, 1995).

Interpretation 1: Large scale quantum computers are possible in principle!

Interpretation 2: If we can control intermediate-scale quantum systems well enough, then we can build large scale universal quantum computers.

Part III: Permanents, determinants,and noise sensitivity of boson sampling• Multiplication is easy Factoring is hard!• Determinants are easy and Permanents are hard!

Both these insights are very old. They were studied by mathematicians well before modern computational complexity was developed. Quantum computers make factoring easy, and also make computing permanents “easier”.

In 1913, Polya proposed the following problem: Show that there is no affixing of ± signs to the elements of the square matrices of order n > 2 such that the determinant of the resulting matrix equals the permanent of the original matrix.

Model 7: Boson Sampling (non interacting bosons)

Troyansky-Tishby (1996), Aaronson-Arkhipov (2010, 2013):

Given a complex n by m matrix X with orthonormal rows. Sample sub-multisets of columns according to the absolute value-squared of permanents.This task is referred to as Boson Sampling.

Quantum computers can perform Boson Sampling on the nose. There is a good theoretical argument by Aaronson-Arkhipov (2010) that these tasks are beyond reach for classical computers.

Boson Sampling (permanents):{1,1} – 0 {1,2} – 1/6 {1,3} – 1/6{2,2} – 2/6 {2,3} – 0 {3,3} – 2/6

Fermion Sampling (determinants):{1,2} – 1/6 {1,3} – 1/6 {2,3} – 4/6

Input matrix

1 2 3

Model 8: Noisy (robust) Boson Sampling

(Kalai-Kindler 2014) Let G be a complex Gaussian n x m noise matrix (normalized so that the expected row norms is 1). Given an input matrix A, we average the Boson Sampling distributions over (1-t)1/2 A + t1/2 G.

t is the rate of noise.

Now, expand the outcomes in terms of Hermite polynomials. The effect of the noise is exponential decay in terms of Hermite degree.

The Hermite expansion for the BosonSampling model is beautiful and very simple! Thank you Catherine

Goldstein

Noise stability/sensitivity of BosonSampling

Theorem 5 (Kalai-Kindler, 2014): When the noise level is constant, distributions given by noisy Boson Sampling are well approximated by their low-degree Fourier-Hermite expansion. (Consequently, these distributions can be approximated by bounded-depth polynomial-size circuits.)

Theorem 6 (Kalai-Kindler, 2014): When the noise level is larger than 1/! noisy boson sampling are very sensitive to noise, with a vanishing correlation between the noisy distribution and the ideal distribution.

The huge computational gap (left) between Boson Sampling (purple) and Fermion Sampling (green) vanishes in the noisy version.

Part IV: The quantum computer puzzle NISQ-systems

The crucial theoretical and experimental challenge is to understand Noisy-Intermediate-Scale Quantum (NISQ) systems.

Major experimental efforts are aimed at demonstrating “quantum supremacy” using pseudo-random circuits and building good quality quantum error correcting codes.

Noise stability/sensitivity of NISQ systems

Conjecture: Both 2014 theorems of Kalai and Kindler extend to all NISQ systems (in particular, to noisy quantum circuits) and to all realistic forms of noise:

1. When the noise level is constant, distributions given by NISQ systems are well approximated by their low-degree Fourier expansion. Hence they represent low level computational class.

2. For a wide range of lower noise levels, NISQ-systems are very sensitive to noise, with a vanishing correlation between the noisy distribution and the ideal distribution. At this range the distribution will depend on fine properties of the noise.

Predictions of near-term experiments based on Kalai-Kindler 2014

For the distribution of 0-1 strings based on a quantum pseudo-random circuit or a circuits for surface code:

a) For a larger amount of noise- you can get robust experimental outcomes but they will represent LDP (Low Degree Polynomials)-distributions which are far-away from the desired noiseless distributions.

b) For a wide range of a smaller amount of noise- your outcome will be chaotic. This means that the resulting distribution will strongly depend on fine properties of the noise and that you will not be able to reach robust experimental outcomes at all.

Predictions of near-term experiments (cont.)

c) The effort required to control k qubits to allow good approximations for the desired distribution will increase exponentially and will fail at a fairly small number of qubits.(My guess ≤ 20.)

d) (Related to my work before 2012) In the NISQ-regime, gated qubits will be subject to errors with large positive correlation.And so will any pair of entangled qubits. This will lead to a strong effect of error-synchronization.

By ICM 2022 and ICM 2026 we will know better

The theoretical argument against quantum computers

The brief version

1. Quantum supremacy requires quantum error correction 2. Quantum supremacy is easier than quantum error-correction

1. and 2. together imply that quantum supremacy is simply out of reach.

Classical computation requires classical error correction, but classical error correction is supported by low level computation!

The theoretical argument against quantum computers

(A) Probability distributions described (robustly) by NISQ devices can be described by law-degree polynomials (LDP).

LDP-distributions represent a very low-level computational complexity class well inside (classical) AC0.

(B) Asymptotically-low-level computational devices cannot lead to superior computation.

(C) Achieving quantum supremacy is easier than achieving quantum error correction.Part (C) fails for classical computation. There is strong theoretical evidence for (A) and empirical and theoretical evidence for (C).

(A) The computational power of NISQ systems

NISQ-circuits are computationally very weak, unlikely to allow quantum codes needed for quantum computers.

What could be the law regarding noise?

? Law 0: Every quantum evolution is noisy. (Violates QM, rejected)

! Law 1: Time dependent quantum evolutions are noisy*! Law 2: Noise (above the level allowing QC) is a necessary

ingredient in modeling local quantum systems*! Law 3: Quantum observables are noise-stable in the

Benjamini-Kalai-Schramm sense.*

*needs mathematical formulation and leads to interesting mathematics.

Open problems

• Prove that the crossing event in 3-D percolation is noise sensitive

• Study noise-stable versions of various models from statistical physics

• Study the math and physics above the fault-tolerance threshold*

• Study noise-stable versions of various models from quantum computing.

• Extend the noise sensitivity/noise stability framework to models where Z/2Z or S1 are replaced by other groups relevant to physics (e.g. SU(2), SU(3)). Explore and explain explicit constants coming from noise stability.

* See the Proceedings paper for some directions and ideas.

Study noise sensitivity/stability for models, in statistical physics, combinatorics, theoretical computer science, quantum physics and game theory.

Conclusion

Understanding noisy quantum systems and potentially even the failure of quantum computers is related to the fascinating mathematics of noise stability and noise sensitivity and its connections to the theory of computing. Exploring this avenue may have important implications to various areas of quantum physics.

Thank you very much!

!הבר הדות

An optimistic peace of information

An amazing scientific partnership between Jordan, Cyprus, Egypt, Iran, Israel, Pakistan, the Palestinian Authority, and Turkey

SESAME (Synchrotron-light for Experimental Science and Applications in the Middle East) is a “third-generation” synchrotron light source that was officially opened in Allan (Jordan) on 16 May 2017.

Additional slide: Important analogies

• The analogy between classical and quantum computers

• The analogy between quantum circuits and BosonSampling

• The analogy between BKS study of noise stability and noise sensitivity, and noisy quantum computation.

• The analogy between surface codes that Google tries to build and topological quantum computing pursued by Microsoft

• The analogy between Majorana fermions in high energy physics and in condensed matter physics.