What can we Learn From a Failure of Quantum Computers

Gil KalaiEinstein Institute of Mathematics

APS - Racah Institute of physics workshop

Quantum Computing: Achievable Reality or Unrealistic Dreams?Hebrew University of Jerusalem,

January 6, 2015

Background

Quantum Computers

Quantum computers are hypothetical devices based on quantum physics that can out-perform classical computers.

In this lecture we will discuss two directions

toward quantum computation: • Universal quantum circuits, and • BosonSampling.

Qubits and quantum computers

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.

The memory of a quantum computer (quantum circuit) consists of n qubits. The state of the computer is a unit vector in the tensor product of all the 2-dimensional Hilbert spaces corresponding to the qubits. We can perform ``gates'' on one or two qubits. There is a small list of gates needed for universal quantum computing.

A gate is a unitary transformation acting on thecorresponding 2- or 4-dimensional Hilbert space.

The state of the entire computer can be measured and this gives a probability distribution on 0-1 vectors of length n.

BosonSampling

Tishby and Troyansky (1996) and Aaronson and

Arkhipov (2013):

Manipulation of n noninteracting bosons will allow sampling n by n submatrices of a prescribed n by m matrix according to the value of the permanent.

Computational complexity

Quantum Supremacy

Quantum computers will enable us to perform certain computations hundreds of magnitude of order faster than digital computers.

This feature, coined “quantum supremacy” by John

Preskill, could be manifested by experiments in the near future.

Noise

The main concern from the start was that quantum systems are inherently noisy; we cannot accurately control them, and we cannot accurately describe them. To overcome this difficulty, a theory of quantum fault-tolerance based on quantum error-correction was developed.

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.

What will really happen

For every implementation of universal quantum circuits (or of quantum error-correcting codes) the noise level per qubit will scale up with the number of qubits. this will make quantum fault-tolerance impossible.

Every implementation of BosonSampling will fail for a handful of bosons much before any quantum supremacy is demonstrated.

The scientific challenge

To demonstrate noise modeling supporting these

assertions; To offer predictions, both conceptual and of

computational nature, based on principles of no quantum-supremacy and no quantum-fault-tolerance.

Reasons to disbelieve

Reasons to disbelieve: How quantum computers will

change reality• A universal machine for creating quantum states and

evolutions could be built.• Complicated states and evolutions never encountered

before could be created• States and evolutions could be constructed on arbitrary

geometry• Emulated quantum evolutions could always be time-

reversed• The noise will not respect symmetries of the state • Fantastic computational complexity consequences.

Reasons to disbelieve 2The Magnitude of improvement

Quantum computers represent enormous unprecedented order-of-magnitude improvement for controlled physical phenomena as

well as for algorithms.

Seth Lloyd: “We can build computers that can do computations that no classical computer can do even if it is of the size of the entire universe.”

Reasons to disbelieve 2The Magnitude of improvement

Quantum computers represent enormous unprecedented orders-of-magnitude improvement for controlled physical phenomena as

well as for algorithms. Nuclear weapons ~ 7 Telegraph ~3 (very charitably < 10), Meridor’s claim 4-5 Computer memories < 12 Eratosthenes sieve < 5; Modern algorithms for factoring < 6 Modern algorithms for primality < 5; Combined effect of linear

programming breakthroughs < 6 BosonSampling with 100 bosons > 100 Quantum factoring > 100

Reasons to disbelieve 3Physics computations

QED computations allow to describe various physical quantities in terms of a power series

∑ ck αk

Where ck is the contribution of k-loops Feynman diagrams and α

is the fine structure constant (around 1/137). Quantum computers will (likely*) allow to compute these terms

and sums for large values of k with hundred digits accuracy even in regimes where they have no physical meaning.

(*This motivated QC to start with, is supported by recent work of Jordan, Lee, and Preskill, and is often taken for granted.)

Reasons to disbelieve 4BosonSampling

• Physics reason: You are “hitting” a state in an n(m-n)-dimensional variety inside a relevant Hilbert space of dimension mn (Your non-interacting bosons have huge degree of freedoms coming from weak interactions and from mode mismatches)

• Computational complexity reason: (Kalai Kindler ‘14) BosonSampling devices represent (robust-to-noise) bounded depth computation

BosonSampling represents Bounded Depth Computation

Computational complexity reason: (Kalai Kindler ‘14) BosonSampling devices represent (robust-to-noise) bounded depth computation

P

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Predictions from the failure of quantum computers

My Basis hypothesis

• Quantum supremacy requires quantum fault-tolerance

• Quantum fault-tolerance is not possible.

• The failure of quantum fault-tolerance and quantum supremacy can serve as a powerful conceptual and computational tool in quantum physics.

A refined hypothesis

• 1) Any form of computation (beyond “robust-to-noise bounded depth computation”) requires fault-tolerance.

• 2) Every physically feasible mechanism for fault-tolerance is based on a repetition (averaging) mechanism which, allows only classical information and computation.

A basic assumption

For general quantum systems, there are systematic relations between the noise and the (entire) quantum evolution.

For implementations of quantum circuits there are systematic relations between the target state and the noise.

Important consequence: Noise modeling is implicit.

Predictions for implementations of quantum circuits

1.1 Two-qubits behavior. Any implementation of quantum circuits is subject to noise for which errors for a pair of entangled qubits will have substantial positive correlation.

1.2 Error-synchronization. For complicated target states highly synchronized errors will occur.

1.3 Error-rate. For complicated evolutions (or for evolutions approximating complicated states) the error rate (in terms of qubits-errors) scales up with the number of qubits.

1.4 No encoded qubits. Encoded qubits cannot be stable. (They cannot be substantially more stable than the raw qubits used for constructing them.)

Surface codes implementations

There are several groups attempting to create stable encoded qubits based on surface codes applied to superconducting qubits. These attempts will bring my conjectures to test. For example, John Martinis proposes to create, in a few years, encoded qubits based on distance-five surface codes, where each encoded qubit depends on about a hundred raw qubits. My conjectures predict that very positively-correlated errors for individual raw qubits will emerge, leading also to error rate scaling up linearly with the number of raw qubits. In particular, all these attempts of creating much more stable logical qubits based on surface code will fail.

Predictions for general open quantum systems

2.1 rate. For a noisy quantum system a lower bound for the rate of noise in a time-interval is a measure of non-commutativity for the projections in the algebra of unitary operators in that interval.

2.2 Smoothed Lindblad. Noisy quantum evolutions are subject to noise with a substantial correlation with smooth Lindblad evolutions.

2.3 Bounded depth. Noisy bounded-depth polynomial-size quantum computation is the limit for quantum states achievable byany implementation of quantum circuits

The noise depends on the future evolution!

There is a systematic relation between the law for the noise at a time and the entire evolution (including the future evolution)

It is not that the evolution in the future causes the behavior of the noise in the past but rather the noise in the past leads to constraints on feasible evolutions in the future.

The Noise depend on the future evolution!

Such dependence occurs also for classical systems. Without refueling capabilities, the risk of space missions in take-off strongly depends on the details of the full mission. (Such dependence can be eliminated with refueling capabilities).

A principle of no quantum fault-tolerance implies that in the quantum setting such dependence cannot be eliminated.

Cooling

• Within a symmetry class of quantum states (or for class defined in a different way) the bounded-depth requirement provides an absolute lower bound for cooling.

• Stable anyonic qubits, and low-temperature anyonic states cannot be constructed.

Fluctuation

Fluctuations in the rate of noise for interacting N-elements systems(even in cases where interactions are weak and unintended) scale like N and not like √N .

Non-interacting bosons (BosonSampling)

Robust experimental outcomes for systems of non-interacting bosons can be approximated by low-degree Hermite polynomials and this gives an effective tool for computations.

Two proposed extensions: 1. In wide contexts, robust (or noise-stable) quantum physics experimental outcomes are computationally feasible and stability to noise can lead to effective computational tools.

2. For varietal quantum evolutions and states (those described by a low dimensional algebraic variety in a large Hilbert space), robust experimental outcomes can be approximated by low degree polynomials on the tangent planes.

Classical simulation

Computations in quantum physics can, in principle, be simulated on a digital computer.

• Caveat: 1) Computational shortcuts will require knowing internal parameters of the process which are not available to us (but are available to nature). This can be seen as the learnability-gap in computational complexity. It is computationally hard to learn functions of even low level computational-complexity class.

• 2) Heavy computations can occur for a model representing a physical process that depends on much more parameters than represented by the input size.

Classical simulation: Caveats and Prospects

• 3) An effective efficient model can actually be much more complicated to represent than a simple non-efficient model that agrees with it on physically relevant inputs.

• 4) And, of course, heavy computation can occur when we simply do not know the correct model or relevant computational tool.

Even with all these caveats the prediction about classical simulability is powerful. There are quite a few examples of computations from quantum physics where apparently superior computational power is needed. We witness robust physical behavior of fairly complicated systems and we witness larger and larger computational power needed to allow predictions that fits experiment. We need to study these cases one by one.

Other connections

A principle of no quantum fault-tolerance may �shed light on familiar issues and controversies in quantum physics, and may enable to capture into the scientific grounds some foundation uncharted territories.

Speculative connection: The emergence of locality; the emergence of specetime; the firewall blackhole paradox; The measurement problem; QM and free will;…

Two Slogans

The importance of quantum fault-tolerance to physics is similar to the importance of non-deterministic computation in the theory of computing – their importance is that they cannot be achieved.

Spacetime is enabled by the failure of quantum fault-tolerance.

Summary Quantum supremacy and quantum fault-tolerance

represent a major phase-transition for noisy quantum systems. Finding the mathematical tools to model and understand the nature of this phase transition, the quantum fault-tolerance barrier, is important to the understanding of open quantum systems, quantum thermodynamics, and approximations in quantum physics. A principle of ``no quantum supremacy'' will have major conceptual and computational consequences, in quantum physics.

תודה רבה

Smoothed Lindblad (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].

Important to remember: Quantum computers represent a counterfactual

situation Since universal quantum computers are hypothetical some insights

for why they might fail are counter-factual and to become interesting they should be extended to quantum systems without superior computing capabilities. As Greg Kuperberg puts it:

“If a car plunges over a cliff, you don’t necessarily need a careful model of every rock that flies through the windshield.”

Quantum devices that demonstrate superior quantum computing are like cars that plunge over a cliff. We need to understand and model how cars behave in ordinary circumstances before they plunge over the cliff.

The emergence of locality Locality means (on the combinatorial side) that quantum

interactions are limited to very few particles (qubits) and (on the geometric side) that those involved particles are confined geometrically. Enforcing local rules on the nature of noise (approximations) allows highly non-local behavior for controlled systems via quantum fault-tolerance. I predict that the various conjectures on the nature of noise, will lead to “locality” being emerged both for the law for the noise/approximation and for the law for the approximated evolution.

But is it fundamental? Does the failure of quantum computing/fault-

tolerance represent fundamental physics obstacle?

(Rather than a practical engineering one)

Short answer: As a mathematician it does not matter. Quantum fault-tolerance and quantum supremacy represent a major phase transition that requires mathematical modeling and understanding.

Long answer: Yes!