Merlin-Arthur%20Games%20and%20Stoquastic%20Hamiltonians

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Merlin-Arthur Games and Stoquastic Hamiltonians

• B.M. Terhal, IBM Research
• based on work with Bravyi, Bessen, DiVincenzo
  Oliveira.
• quant-ph/0611021 quant-ph/0606140

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Quantum Complexity Theory

• Heuristic methods have been developed to solve
  classical problems (simplex method for linear
  programming, simulated annealing, Monte Carlo
  methods).
• Classical complexity theory aims at making
  rigorous what problems can be solved efficiently
  and which ones are hard (and how hard they are).
• Physicists have developed (heuristic) methods to
  solve quantum problems (perturbation theory, WKB
  approximation, mean-field theory etc. etc.)
• Quantum complexity theory aims at showing which
  problems a classical machine can solve
  efficiently, which ones a quantum computer can do
  better, and which ones are hard and how hard.
• The how hard-aspect is sometimes characterized in
  terms of interaction of an ordinary human being
  (Arthur) with an all-powerful prover (Merlin).

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NP
Success story of complexity theory in physics
the notion of NP and NP-completeness. Is the
question P?NP a mathematics or physics
question A mathematical question with physical
consequences, a universe where PNP would
probably be much different. Useful notion of
NP-completeness.
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Satisfiability
The k-SAT problem Let x be an n-bit string. Let
Ci be a constraint that depends only on the value
of some k bits of x. A bitstring x can either
satisfy or not satisfy the constraint Ci
depending on the value of these k bits. Take a
set of constraints Ci for i1 to m with
mpoly(n). Q Is there a bit string that
satisfies all m constraints? If yes, a prover
can give you the bitstring and you can check the
m constraints. If no, no prover can convince you
that there is such bitstring. k-SAT is
NP-complete
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Beyond NP
What is the physical relevance of these
classes? What are the complete-problems?
QMA (or Quantum NP)
NP
MA
Stoquastic MA
MA probabilistic version of NP QMA quantum
version of NP
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Definitions
Interaction between Arthur and Merlin Arthur runs Arthur runs V(..,x)
NP Proof y by Merlin Polynomial time algorithm (in x) with input y If Yes, there exists a y, V(y,x)1 If No, for all y, V(y,x)0
QMA Quantum proof ? by Merlin Poly-time quantum algorithm If Yes, there exists a state ?, Prob(V?,x) ? 2/3 If No, for all states ?, Prob(V,?,x) ? 1/3
MA Proof y by Merlin Poly-time probabilistic algorithm with input y If Yes, there exists a y, Prob(Vy,x)1 If No, for all y Prob(Vy,x) ? 1/3
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The Story

• The Quantum Ising Spin Glass problem is complete
  for QMA.
• Note that QMA allows for an error if the answer
  is yes (if yes, Arthur decides yes with
  probability at least (say) 2/3). Note that NP and
  MA
• do not have this error (MA allows for error in
  no-instances)
• For the class QMA1 in which Arthur decides Yes
  with probability 1,
• the complete problem is the quantum
  satisfiability problem.
• For MA the complete problem (first known
  MA-complete problem) is the stoquastic
  satisfiability problem. Relation to stoquastic
  Hamiltonians and classical satisfiability
  problem.
• For stoquastic MA (weird class) the complete
  problem is the
• Stoquastic Ising Spin Glass problem.
• Beliefs in computer science MANPStoquastic
  MAAM

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Quantum Ising Spin Glass

• Quantum Ising Spin Glass (QISP)
• Given is a k-local Hamiltonian H?i Hi on n
  qubits and
• given ? lt ? (with ?-? ? 1/poly(n)). Hi ?
  poly(n). We are promised that
• the ground-state energy ?(H)) is either ? ? or ?
  ?.
• Q Is there a state with energy less than ??
• k-local means that the Hamiltonian is a sum of
  terms each of which acts
• non-trivially on at most k qubits.
• Kitaev 5-local QISP is QMA-complete.
• Strongest results to date 2-local QISP on qudits
  of a 1-D chain is
• QMA-complete (Aharonov,Gottesman, Kempe, Irani).
• Quantum Ising Spin glass is also called the local
  Hamiltonian problem.

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Quantum k-SAT
Ground-state of Hamiltonian is not always the
one which is the ground-state of each individual
term in the Hamiltonian. frustration free
property So a different question to ask is is
there a state which is the ground-state of each
local term in a quantum Ising spin glass
Hamiltonian? Such state satisfies all local
constraints. We can always shift the Hamiltonian
such that the energy of this satisfying state is
0. Then the satisfying state has the property
of being in the null-space of a bunch of k-local
projectors. Quantum k-SAT on n qubits Given a
set of k-local projectors Pi on n qubits and ? ?
1/poly(n). Either - there is a state that is in
the space spanned by all projectors Pi.
(yes-instance), - or any state ? there exists
an i such that lt? Pi ?gt ? 1-?
(no-instance). (Note we are switching null-space
and 1 space here)

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Connection with classical k-SAT

Classical k-SAT problem can be viewed as quantum
k-SAT where the projectors are diagonal in
computation basis. Some bit-strings in 1 space,
some in null-space. Promise on no-instances is
not needed. There is something in between
diagonal k-local projectors and general k-local
projectors.the stoquastic k-SAT problem
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Stoquastic Hamiltonians
Physicists have known for a long time that not
all Hamiltonians are created equal. Some are more
quantum than others. Physics lingo Avoid the
sign-problem, Quantum ground-state problem
mapped onto classical partition function
problem, Greens function Monte Carlo
techniques What is exactly the property of
these Hamiltonians and what does this feature do
to the complexity of the lowest-eigenvalue
problem, the power of these Hamiltonians for
implementing an adiabatic quantum computer, etc?
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Stoquastic Hamiltonians
General definition stoquastic Hamiltonians are
real and have non-positive off-diagonal elements
in some standard (product) basis igt. Then
GI-t H for some (real) t is a non-negative
matrix (in this basis) or consider the
nonnegative Gibbs matrix Ge-bH . Perron-Frobenius
Eigenvector with largest eigenvalue of G
(ground-state of H) has the property that ?gt?i
?i igt where ?i ? 0. P(i)?i/?i ?i is a
probability distribution. Our excuses for the
word stoquastic.introduced not without reason.
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Examples of Stoquastic Hamiltonians
Particles in a potential Hamiltonian is a sum
of a diagonal potential term in position xgt and
off-diagonal negative kinetic terms
(-d2/dx2). All of classical and quantum
mechanics. Quantum transverse Ising model
Ferromagnetic Heisenberg models (modeling
interacting spins on lattices) Jaynes-Cummings
Hamiltonian (describing atom-laser interaction),
spin-boson model, bosonic Hubbard models,
Bose-Einstein condensates etc. D-Waves Orion
quantum computer Non-stoquastic are typically
fermionic systems, charged particles in a
magnetic field. Stoquastic Hamiltonians are
ubiquitous in nature. Note that we only consider
ground-state properties of these Hamiltonians.
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Local Stoquastic Hamiltonians

• Remember k-local Hamiltonians are those that can
  be written as a
• sum of terms acting non-trivially on k qubits
  (qudits).
• Our results hold for local term-wise stoquastic
  Hamiltonians, each
• local term is stoquastic.
• Side Remark
• Local termwise-stoquastic is not necessarily the
  same as local stoquastic
• 2-local on qubits termwise-stoquastic is
  stoquastic.
• 3-local on qubits there is a counter-example of
  a Hamiltonian which is
• stoquastic but not term-wise stoquastic (also
  allowing 1-local unitary
• transformations).

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Stoquastic k-SAT
Quantum k-SAT remember we have a collection of
projectors Pi (mpoly(n) of them). Now we demand
that Pi are projectors with nonnegative entries
in the computational basis. Thus H?i (I-Pi) is
stoquastic. (in fact termwise-stoquastic, i.e.
each Hi is stoquastic). How to prove that this
problem is a complete problem for the classical
class MA? Sketch of ideas Completeness for a
problem Q 1. Prove that a problem Q is in the
class. 2. Prove that any problem in the class
can be reduced to Q (if we can solve Q we can
solve any problem in the class)
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Stoquastic k-SAT
Completeness for a problem Q 1. Prove that a
problem Q is in the class. 2. Prove that any
problem in the class can be reduced to Q (if we
can solve Q we can solve any problem in the
class) Kitaev showed 2. for the quantum Ising
spin glass by converting Arthurs quantum
verifying circuit into a Hamiltonian. We do 2.
by treating a probabilistic verifying circuit as
a restricted quantum circuit and applying
Kitaevs construction to obtain a restricted
Hamiltonian. This Hamiltonian turns out to be
stoquastic! In MA, if answer is yes, verifying
circuit with proof always outputs yes (zero
error) This results in a Hamiltonian where the
ground-state (in the yes-case) satisfies each
local constraint, i.e the stoquastic k-SAT
problem.
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Containment in MA
. Given a set of k-local projectors Pi
(mpoly(n) of them) with nonnegative entries in
the computational basis and ? ?
1/poly(n). -Yes-instance there is a state that
is in the space spanned by all projectors Pi.
- No-instance for any state ? there exists an
i such that lt? Pi ?gt ? 1-? Consider the
nonnegative (symmetric) matrix G m-1 ?i Pi. -
Yes-instance the largest eigenvalue of G is
1. - No-instance the largest eigenvalue of G is
less than 1-?/m

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Containment in MA

The idea of the proof Merlin provides starting
point to a random walk on n bit strings. The
random walk is a true random walk in case of a
Yes-instance. Arthur will test whether the
transition probabilities during the walk sum up
to 1 and do a final test (which depends on the
starting point). In case of a no-instance there
is a high probability that some test fails. If a
test fails, Arthur decides No.
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MA (versus NP)
Special version of stoquastic k-SAT problem can
be viewed as a bunch of constraints (like SAT)
and relations. Graph with vertices which are
n-bit strings. Vertex x is good when bit-string
is in the support of all projectors. We have an
edge between vertex x and x when there exists an
i such that ltxPixgt gt 0. Graph has degree m (is
poly(n)) Special Stoquastic k-SAT problem that
is still MA-complete Does the graph contain a
disconnected sub-graph of only good vertices?
How to prove that there is such sub-graph when
sub-graph can be exponentially large? Walk
around on sub-graph starting from point provided
by prover. Promise guarantees that in the no-case
after some poly(n) time you will get to a bad
vertex. Note that you would explore only a set
with poly(n) vertices (chosen by random walk)

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Discussion
Adiabatic evolution with general (local)
Hamiltonians is universal for quantum
computation. What is the power of adiabatic
quantum computation with stoquastic (local)
Hamiltonians? Can we classically simulate it? Do
heuristic methods/approximation algorithms work
better for stoquastic Hamiltonians? Problem is
still hard but more classically hard.