Transcript What can we Learn From a Failure of Quantum Computers Gil Kalai Einstein Institute of Mathematics APS - Racah Institute of physics workshop Quantum Computing: Achievable Reality.

What can we Learn From a Failure
of Quantum Computers
Gil Kalai
Einstein 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 outperform 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 4dimensional 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 errorcorrection 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-faulttolerance.
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 timereversed
• The noise will not respect symmetries of the state
• Fantastic computational complexity consequences.
Reasons to disbelieve 2
The Magnitude of improvement
Quantum computers represent
enormous unprecedented order-ofmagnitude 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 2
The Magnitude of improvement
Quantum computers represent enormous unprecedented orders-ofmagnitude 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 3
Physics 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 4
BosonSampling
• Physics reason: You are “hitting” a
state in an n(m-n)-dimensional
variety inside a relevant Hilbert
space of dimension mn (Your noninteracting 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
AC0
Predictions from the failure of
quantum computers
My Basis hypothesis
• Quantum supremacy requires quantum faulttolerance
• 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-tonoise bounded depth computation”) requires
fault-tolerance.
• 2) Every physically feasible mechanism for faulttolerance 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 noncommutativity 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 polynomialsize 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 boundeddepth 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 Nelements 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 noisestable) 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 nondeterministic 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 faulttolerance. 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/faulttolerance 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!