1. The Quantum Computer Puzzle: Lecture 3: what can we learn from a failure of quantum computers
Gil Kalai
Einstein Institute of Mathematics
Hebrew University of Jerusalem
Distinguished Lecture Series
Department of Mathematics
Indiana University Bloomington
Fall 2016

2. The quantum computer puzzle
Quantum computers are hypothetical devices, based on quantum physics, which would enable us to perform certain computations hundreds of orders of magnitude faster than digital computers. The question whether quantum computers are feasible is an important clear-cut scientific/technological question.
My expectation for a negative answer is based on the following principle:
One cannot engineer a computationally inferior device to perform a computational superior task.
This refers to direct demonstrations of quantum supremacy for small number of elements as well as to the creation of quantum error correcting codes necessary for quantum fault-tolerance.

3. Lecture 1: Classical and quantum efficient computation
What is computation? What is a qubit? What is a quantum computer?
A qubit is a piece of quantum memory. The state of a qubit can be described by a unit vector in a 2-dimensional complex Hilbert space H.

4. From lecture 1: Noise
Noise refers to the general effect of neglecting degrees of freedom. The study of noise is relevant not only to controlled quantum systems but to many other aspects of quantum physics.
There are two mathematically equivalent ways to describe noisy quantum systems: One way is to consider a larger Hilbert space which accounts for the neglected degrees of freedom. The second way is to consider mixed states and quantum operations.
For our purposes it is enough to consider depolarizing noise.

5. The Optimistic Hypothesis
The engineering effort required to obtain a bounded error level for universal quantum circuits increases moderately with the number of qubits.
It is therefore possible to realize universal quantum circuits with a small bounded error level regardless of the number of qubits.
Thus, large-scale fault-tolerant quantum computers are possible.

6. The Pessimistic Hypothesis
The engineering effort required to obtain a bounded error level for any implementation of universal quantum circuits increases (at least) exponentially with the number of qubits.
Therefore, for every realization of universal quantum circuits the error rate scales up (at least) linearly with the number of qubits
These difficulties will be demonstrated for small quantum systems with handful of qubits.
Thus, quantum computers are not possible.

7. Lecture 2: The theory of noise sensitivity and stability and the anomaly of majority
The majority function admits excellent approximation with low degree polynomials (noise stability) and yet allows the emergence of robust classical information (large influence/sharp threshold/aggregation of information).

8. BosonSampling
Manipulation of n noninteracting bosons will allow sampling n by n submatrices of a prescribed n by m matrix according to the values of the permanent.
BosonSampling was introduced by Troyansky and Tishby in 1996 and was intensively studied by Aaronson and Arkhipov, who offered it as a quick path for experimentally showing that quantum supremacy is a real phenomenon.

9. Sampling via free bosons
Consider the input matrix The output for BosonSampling is a probability distribution according to absolute values of the square 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.

10. BosonSampling is computationally hard for classical computers
Aaronson and Arkhipov proved (2011) that a classical computer with access to random bits cannot perform BosonSampling unless the polynomial hierarchy collapses! FermionSampling, which is sampling according to value of determinants, can be efficiently done with a classical computer.

11. My work with Guy Kindler

12. Concrete computations

14. Noisy BosonSampling
Theorem 1 (Kalai Kindler ‘14): When the noise level is constant, BosonSampling distributions are well approximated by their low-degree Fourier–Hermite expansion. Consequently, noisy BosonSampling can be approximated by bounded-depth polynomial-size circuits. Theorem 2 (Kalai Kindler ‘14): (Kalai and Kindler). When the noise level is 𝜔(1/𝑛) (and 𝑚 ≫ 𝑛2), BosonSampling is very sensitive to noise, with a vanishing correlation between the noisy distribution and the ideal distribution.

15. BosonSampling meets reality
Noisy BosonSampling and Noisy FermionSampling
BosonSampling and FermionSampling

16. The noise models

17. Interpretation
Theorems 1 and 2 give evidence against expectations of demonstrating “quantum supremacy” via BosonSampling: experimental BosonSampling represents an extremely low-level computation, and there is no precedence for a “bounded-depth machine” or a “bounded-depth algorithm” that gives a practical advantage, even for small input size, over the full power of classical computers, not to mention some superior powers.
They also suggest that noise-stable bosonic states could be experimentally demonstrated!

19. Noise stability and low degree polynomials – computations and the physical world.
Low degree polynomials (LDP):
Allow efficient learning;
Low degree polynomials allow classical information and computation (but not quantum error correction);
Low degree polynomials allow (probably) reaching efficiently ground states;
LDP

21. Emergent modeling for noise from the pessimistic hypothesis

22. Basis premises for modeling noise in the larger scales
Modeling is implicit (like PDE’s are..)
There are systematic relations between the noise and the entire evolution.

23. Two qubits behavior
Two-qubits behavior. Any implementation of quantum circuits is subject to noise for which errors for a pair of maximally entangled qubits (cat states) will have substantial positive correlation.

24. Time-Smoothed evolutions (discrete time)
We consider a quantum circuit that runs for T computer cycles, we let Ut denote the intended unitary operator for the t-th step, and we start with a noise operation Et for the t-step.
Then we consider the noise operator
Where Us,t denotes the intended unitary operation between step s and step t. (t can be larger or smaller than s) K is a positive kernel defined on [-1,1].

25. Predictions from the pessimistic hypothesis

26. Teleportation, superposition, predictability, time-reversing…
Quantum states within and near the threshold cannot be teleported.
Quantum states within and near the threshold cannot be superposed.
Quantum states within and near the threshold cannot be implemented on an arbitrary geometry.
Some quantum evolutions within and near the thresholds cannot be time reversed.
Some quantum evolutions beyond the threshold cannot be predicted.

27. Predictions on living cats
There are quantum states that can be created but cannot be teleported; Two states that can be created separately but cannot be superposed; States that cannot be implemented on an arbitrary geometry.
Some quantum evolutions cannot be time reversed and cannot be predicted.
Cats could not be teleported; it will be impossible to reverse the life-evolution of the cat, it will be impossible to implement a cat on a device with very different geometry, to superpose the life-evolutions of two distinct cats, and, even the cat is placed in an isolated and monitored environment, its life-evolution cannot be predicted.

28. Summary
The remarkable progress witnessed during the past two decades in the field of experimental physics of controlled quantum systems places the decision between the pessimistic and optimistic hypotheses within reach.
I expect that the pessimistic hypothesis will prevail, yielding important outcomes for physics, the theory of computing, and mathematics.
Our journey through the theory of noise stability and noise sensitivity, probability distributions described by low-degree polynomials and implicit modeling for noise, may provide some of the pieces needed for solving the quantum computer puzzle.

29. Thank you!