Research on Quantum Algorithms
at the Institute for Quantum Information and Matter
J. Preskill, L. Schulman, Caltech
preskill@caltech.edu / www.iqim.caltech.edu/
Objective
• Improved rigorous estimates of thresholds for
fault-tolerant quantum computation.
• Quantum algorithms beyond the hidden
subgroup paradigm.
• Quantum and classical simulation methods for
quantum many-body systems.
• New approaches to physically robust quantum
computation.
Dissipative preparation of topological states.
Objective Approach
• Quantum algorithms for simulating local quantum
systems.
• Novel applications of the quantum Fourier
transform and other transforms.
• Customizing quantum fault tolerance for physically
motivated noise models.
• Schemes for physically robust quantum storage and
processing.
• Characterizing Hamiltonian complexity.
• Quantum-resistant classical cryptography.
Status
•
•
•
•
•
Quantum algorithms for simulating particle
collisions in fermionic quantum field theories.
Optimal algorithms for dissipative preparation of
topological code states.
Characterization of fault-tolerant logical
operations in topological subsystem codes.
Proof of rapid thermal mixing for graph state
Hamiltonians and free fermions.
Stable, geometry-preserving discretizations of
arbitrary compact groups.
Research on Quantum Algorithms
at the Institute for Quantum Information and Matter
J. Preskill, L. Schulman, Caltech
preskill@caltech.edu / www.iqim.caltech.edu/
• Progress on last year’s objectives – FY13-14
- Quantum algorithms for simulating particle collisions in fermionic quantum field theories.
- Results on optimal contractions of generic tensor networks, with applications to critical systems.
- Optimal algorithms for dissipative preparation of topological code states.
- Characterization of fault-tolerant logical operations in topological subsystem codes.
- Evidence for probability distributions that can be sampled efficiently quantumly but not classically.
- Asymptotically uniform, stable, geometry-preserving discretizations of arbitrary compact groups.
- Properties of alpha-Renyi generalizations of quantum conditional mutual information.
- New framework for quantum-proof pseudorandomness based on operator space theory.
- Proof of rapid thermal mixing for graph state Hamiltonians and free fermions.
• Research plan for the next 12 months – FY14-15
-
Show that quantum field theory simulation solves a hard (BQP-complete) problem.
Characterize the logical operators that can be performed using constant-depth unitaries in TQFT.s
Stronger impossibility proofs for quantum obfuscation schemes.
Relate quantum-proofness of randomness condensers to Bell inequality violation.
Show that quantum messages and entanglement increase the power of multi-prover interactive proofs.
Improved rigorous bounds on quantum memory times.
• Long term objectives
- Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for
protecting quantum systems from noise.
- Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.
Research on Quantum Algorithms at the IQIM
Faculty:
John Preskill
Alexei Kitaev
Leonard Schulman
Gil Refael
Thomas Vidick
Faculty Associates:
Todd Brun
Daniel Lidar
Steven van Enk
Kirill Shtengel
Sandy Irani
Postdocs:
Gorjan Alagic
Spiros Michalakis
→ Copenhagen
Fernando Pastawski
Mario Berta
Kristan Temme
Andrew Essin
Ling Wang
Glen Evenbly
Beni Yoshida
→ UC Irvine
Olivier Landon-Cardinal
Students:
Michael Beverland
Bill Fefferman → U. Maryland
Matt Fishman
Alex Kubica
Shaun Maguire
Evgeny Mozgunov
Sujeet Shukla
Nicole Yunger-Halpern
Undergrads (6 in 2014)
Visitors:
Many
Postdocs arriving 2014-15:
Omar Fawzi (ETH)
David Gosset (Waterloo)
Stacey Jeffery (Waterloo)
Simulating quantum evolution
-- Goal: simulating real-time evolution of many-particle
systems with better accuracy than is classically achievable.
-- Simulating quantum chemistry.
-- Simulating quantum matter.
-- Simulating quantum field theory.
-- Simulating quantum gravity.
-- AdS/CFT, BFSS Matrix model, etc.
-- Thinking about how to simulate quantum field theory on
a classical computer elucidated the foundations of field
theory. Will thinking about how to simulate string theory
on a quantum computer help us answer: What is string
theory?
Jordan, Lee, Preskill
Quantum algorithms for quantum field theories
-- Feynman diagrams have limited precision,
particularly at strong coupling.
-- Classical lattice methods can compute
static properties, but cannot simulate
dynamics
A quantum computer can simulate particle collisions, even at high energy
and strong coupling, using resources (number of qubits and gates) scaling
polynomially with precision, energy, and number of particles.
-- Estimate errors due to regulating (spatial lattice and approximating
continuous variable fields by qubits).
-- Efficient procedure for preparing (strongly-coupled) vacuum and initial
wave packet states, simulating time evolution, measuring final state.
Does the quantum circuit model capture the
computational power of Nature?
What about quantum gravity?
Simulating quantum field theory
Jordan, Lee, Preskill
Input: a list of incoming particle momenta (particles are actually wave
packets with some momentum spread).
Output: a list of outgoing particle momenta.
Goal is to sample accurately from the distribution of final state particles that
would be produced in a high energy collision in a (strongly coupled) field
theory.
Previous work: Consider a self-coupled scalar field in d = 1, 2, 3, spatial
dimensions. Digitize field at each lattice point using nb qubits, where nb
scales logarithmically with energy and accuracy.
Procedure:
(1) Prepare free field vacuum.
(2) Prepare free field wavepackets.
(3) Adiabatically turn on the coupling constant (t).
(4) Evolve for time T using interacting Hamiltonian H.
(5) Adiabatically turn off coupling
(6) Measure field modes of free theory.
Need to discretize the problem, and keep track of resulting errors.
Simulating quantum field theory
Need to discretize the problem, and
keep track of resulting errors.
Errors arise from, e.g.
-- digitization of field.
-- lattice spacing a. Error is O(a2).
-- Trotter step size.
-- diabatic particle production
during state preparation.
Jordan, Lee, Preskill
Example: self-coupled scalar field in
d = 2 spatial dimensions.
Scaling with error dominated by
preparation of (free) vacuum:
G  O  (1/ a 2 ) 2.376   O  (1/ ò) 2.376 
Scaling with energy:
G  O  E6 
Factor of E from Trotter step, E2 from
a ~ E-1, E3 from diabatic effect.
Formally, an infinite number of degrees of freedom per unit volume. But
good approximation with (energy-dependent) lattice spacing.
Even for 2
accuracy.
4 scattering in 1 spatial dimension, ~10,000 qubits for 1%
Can improve E dependence from diabatic particle production somewhat…
How much can we improve scaling with error epsilon? Requires a lattice
theory with a larger lattice spacing a in physical units (RG improvement.)
Simulating fermionic quantum field theory
Jordan, Lee, Preskill
arXiv:1404.7115
Input: a list of incoming particle momenta (particles are actually wave packets
with some momentum spread).
Output: a list of outgoing particle momenta.
Goal is to sample accurately from the distribution of final state particles that
would be produced in a high energy collision in a (strongly coupled) field
theory.
Completed this year (in progress last year): Consider a self-coupled fermionic
field in d = 1 spatial dimensions (e.g., Gross-Neveu model).
Procedure:
(1) Prepare uncoupled fermion modes.
(2) Adiabatically turn on nearest neighbor coupling between modes.
(3) Adiabatically turn on the coupling constant (t).
(4) Excite spatially localized wave packets with time-dependent sources.
(5) Measure charge and postselect on detecting one particle.
(6) Evolve for time T using interacting Hamiltonian H.
(7) Nondestructively measure energy and momentum of outgoing particles.
Need to discretize the problem, and keep track of resulting errors.
Simulating fermionic quantum field theory
Jordan, Lee, Preskill
arXiv:1404.7115
Free fermion vacuum is not Gaussian – prepare it by adiabatically turning on
nearest neighbor coupling between modes.
Fermi minus sign: Use Bravyi-Kitaev encoding at cost O(log L). When a
fermionic gate is applied, relative sign of |0> and |1> depends on occupation
numbers of other modes (e.g. the number of occupied modes to the left of the
given site). We could represent fermion operators as (Jordan-Wigner) nonlocal
string operators at cost O(L), or we could store the partial sums of mode
occupation numbers, but then updates have cost O(L). Better: cleverly choose
partial sums which allow computation of (-1)’s in O(log L) and can be updated
in time O(log L).
Exciting wave packets: Modulate source spatially and temporally to match one
particle states. Make the source weak to avoid creating more than one
particle, but it usually produces nothing. Measure and abort if not particle
created (okay for a collision of a constant number of particles).
Advantage over previous method (in which coupling ramps on after
wavepacket created): works for bound states.
Simulating fermionic quantum field theory
Jordan, Lee, Preskill
Procedure:
(1) Prepare uncoupled fermion modes.
(2) Adiabatically turn on nearest neighbor coupling between modes.
(3) Adiabatically turn on the coupling constant (t).
(4) Excite spatially localized wave packets with time-dependent sources.
(5) Measure charge and postselect on detecting one particle.
(6) Evolve for time T using interacting Hamiltonian H.
(7) Nondestructively measure energy and momentum of outgoing particles.
Need to discretize the problem, and keep track of resulting errors.
Cost is dominated by the adiabatic preparation of the vacuum. Adiabaticity
enforces turn-on time
T  O 1/ a 4ò


where a is the lattice spacing and is the error. Using a high
2 1 o (1)
-order Trotter approximation, the number of gates needed is: G  O (TL / a )

The error due to nonzero lattice spacing scales as ~ a, and another factor of 1/
arises from postselection; hence cost scales with error as
G  O  (1/ ò)8 o (1) 
(seems pessimistic!)

Simulating quantum field theory
Future plans:
BQP-completeness of simulating quantum field theory.
Improve cost.
Massless particles (infrared safe observables).
Gauge fields (start with strong coupling limit).
Ground state preparation by cooling.
Nonzero temperature and chemical potential.
Simulate standard model of particle physics in BQP.
Quantum gravity?
BQP-hardness of simulating quantum field theory (in progress)
-- Is simulating an interacting quantum field theory as hard as
any problem a quantum computer can solve efficiently?
-- Formulate the problem in terms of the response of the
vacuum to smooth spatially and temporally modulated source
fields.
-- Adiabatically turn on and populate potential wells, for a
“dual-rail” encoding of qubits.
-- Perform a universal set of quantum gates using phase shifts
due to particle interactions and tunneling between wells.
-- Avoid measurements until the end of the protocol (with
adaptive number operator measurements even bosonic free
fields can simulate a universal quantum computer). Rules out
quantum error correction.
-- Check that it works for a self-coupled massive scalar field.
Simulating quantum field theory
Future plans:
BQP-completeness of simulating quantum field theory.
Improve cost.
Massless particles (infrared safe observables).
Gauge fields (start with strong coupling limit).
Ground state preparation by cooling.
Nonzero temperature and chemical potential.
Simulate standard model of particle physics in BQP.
Quantum gravity?
Quantum obfuscation
An obfuscator O is an algorithm that maps circuits to circuits, and
1. preserves functionality: C and O(C) implement same function;
2. preserves efficiency: |O(C)| < poly(|C|);
3. obfuscates: any attack A on O(C) has a corresponding black-box simulator SC .
Idea: the circuit O(C) is no more useful than a black box with same functionality.
What if we had one? Generically…
• provable software intellectual property protection;
• software patches that don’t reveal the security hole being patched;
• turn any private-key encryption to public-key encryption (public key is O(Enck));
• delegated computation;
• etc.
Unfortunately… no-go: Barak et al. 2001 showed (black-box) obfuscation is impossible
classically.
Quantum obfuscation
An obfuscator O is an algorithm that maps circuits to circuits, and
1. preserves functionality: C and O(C) implement same function;
2. preserves efficiency: |O(C)| < poly(|C|);
3. obfuscates: any attack A on O(C) has a corresponding black-box simulator SC .
What if O is a quantum algorithm and can output a quantum circuit, or a state?
Work in progress [Alagic & Fefferman]:
• proving impossibility when O(C) is a quantum circuit;
• proving impossibility of two-circuit-obfuscation when O(C) is a quantum state.
Two-circuit obfuscation replaces condition 3 above with:
3. (2-circuit) any attack A on a pair (O(C), O(D)) has a black-box simulator SC, D.
…harder to achieve (thus easier to prove impossible.)
Quantum obfuscation
Work in progress [Alagic & Fefferman]:
• proving impossibility when O(C) is a quantum circuit;
• proving impossibility of two-circuit-obfuscation when O(C) is a quantum state.
Proof approach similar to original Barak et al. paper:
• pick a random string and two unitaries U1, U2 that only differ on that string;
• write down equal-length circuits C1, C2 for both;
• write down a “universal circuit” D that runs its input on the secret string;
• design an adversary that will accept the quantum states O(Cj) and O(D) and run
the second one on the first one, thereby discovering j;
• show that any black-box simulator faces an unstructured search over exponentially
large spaces of strings and circuits.
To get a one-circuit proof, show how to patch D and Cj together into a single circuit
(seems to require cloning if O produces states!)
Quantum obfuscation
What about other definitions of obfuscation? Replace 3 with:
3. (best-possible) O(C) leaks the least information among all circuits that compute fC.
3. (indistinguishability) fC1 = fC2 implies O(C1) = O(C2).
Work in progress [Alagic & Fefferman]:
• impossibility results for both of the above, when the guarantees in the definition
are perfect or statistical;
• basic approach: show how such obfuscators would give a mapping from quantum
state distinguishability to quantum operation distinguishability;
• easy: perfect pure-state obfuscation of classical functions would put coNP in BQP;
• harder: any statistical guarantee for unitaries would put coQMA in QSZK;
• analogous classical results collapse the polynomial-time hierarchy (unlikely!)
Classical simulation of Yang-Baxter
What is the Yang-Baxter equation? Recall braid groups…
•
•
•
•
n pegs on top and bottom;
strands connect top pegs to bottom pegs;
continuous deformations are ok;
forms a group.
Simple way to create a unitary representation of this group:
• assign a qudit to each gate;
• pick a 2-qudit gate R;
• assign twist of strands i and i+1 :
• R must satisfy Yang-Baxter equation:
Why care about local unitary representations of the braid group? Anyons!
=1
Classical simulation of Yang-Baxter
When kind of quantum-algorithmic power do we get by braiding in this model?
[Alagic, Bapat (Caltech undergrad) and Jordan TQC ’14]:
• if d = 2, it’s purely classical power;
• this is true for some higher-dimensional solutions.
Approach:
• use a classification of the qubit solutions due to Hietarinta and Dye;
• provide classical algorithms for simulating circuits constructed from braiding using
such a solution;
• show that some of the solution families extend to higher dimensions, and that our
algorithm applies to such solution families;
• algorithm also applies to other (non-braiding) quantum circuit families not
previously known to be classically simulable.
Classical simulation of Yang-Baxter
When kind of quantum-algorithmic power do we get by braiding in this model?
[Alagic, Bapat (Caltech undergrad) and Jordan TQC ’14]:
• if d = 2, it’s purely classical power;
• this is true for some higher-dimensional solutions.
Two categories of solutions:
• solutions which are “almost” Clifford gates: these are amenable to an exact
stabilizer-style simulation algorithm for observables on at most
polylogarithmically-many qubits;
• other solutions are a bit more complicated: we used a path-integral approach with
probabilistic sampling, and achieve polynomial-time classical calculation of circuit
amplitudes
with 1/polynomial precision;
• potential future work: when do such solutions correspond to anyon models, or
yield link invariants? Can we characterize the case d = 3? Is it ever universal?
Complexity, quantum, and topology
[Alagic] Current project with Caltech undergrad Catharine Lo.
Idea:
• take a well-established conjecture in classical computational complexity;
• apply some tricks from quantum algorithms;
• get a statement about topology of 3-manifolds.
Recall:
• a 3-manifold is a space which locally looks like Euclidean 3-space;
• gluing solids sometimes gives 3-manifolds: glue two solid balls together and you
get the 3-sphere (just like two disks glued along boundary give 2-sphere);
• Theorem: can get all 3-manifolds just by gluing smooth solids with handles!
• there’s a choice: what map to use to do the gluing.
Complexity, quantum, and topology
Recall:
• a 3-manifold is a space which locally looks like Euclidean 3-space;
• gluing solids sometimes gives 3-manifolds: glue two solid balls together and you
get the 3-sphere (just like two disks glued along boundary give 2-sphere);
• Theorem: can get all 3-manifolds just by gluing smooth solids with handles!
• there’s a choice: what map to use to do the gluing.
Further:
• the Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by:
gluing map, given as a list
of geometrically local twists
Manifold: glue g-handled
solids along map f
local unitary representation
of the group of all gluing maps
Complexity, quantum, and topology
The Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by:
Manifold: glue g-handled
solids along map f
local unitary representation
of the group of all gluing maps
gluing map, given as a list
of geometrically local twists
Idea one:
• give a straightforward quantum circuit which encodes the number #h of satisfying
assignments of a formula h into a matrix entry of the circuit;
• show that exponentially small precision is sufficient to recover #h;
• use density of representation Ug and Solovay-Kitaev to show that approximating
WRT with exponentially small precision is thus #P-hard.
Complexity, quantum, and topology
The Witten-Reshetikhin-Turaev invariant of 3-manifolds can be defined by:
Manifold: glue g-handled
solids along map f
local unitary representation
of the group of all gluing maps
gluing map, given as a list
of geometrically local twists
Idea two (inspired by work of M. Freedman):
• assume that
(standard in classical complexity theory);
• Theorem: there exist manifold diagrams Mg, f satisfying: every nearby diagram of
same manifold must have genus at least polylog(g).
• “nearby” means poly-many equivalence moves (analogous to Reidemeister);
• why is this true? Notice: can get WRT exactly when space is of poly(n) dimension.
• so if Theorem were false, the list of equivalence moves together with this exact
calculation would be a poly-length classical proof for a #P-hard problem!
Approximation theory on groups
Recall that the Quantum Fourier Transform (QFT) is the basis of almost all known
exponential quantum speedups. Can we find other transforms of this kind?
The QFT is a discrete transform. What about continuous spaces? To even get this idea
off the ground, we need integration; this in turn requires
• a way to discretize the continuous space;
• a sampling theory with good numerical stability.
There are some results of this kind for specific groups and spaces (e.g., finite
symmetric group, SO(3), tori, spheres).
Our goal: develop a generic theory for all compact groups. [Alagic Russell Schulman]
Approximation theory on groups
Results:
Theorem 1. Let G be a compact group and B a finite set of irreducible representations
of G. For every
there is a faithful -uniform quadrature rule for Bband-limited functions of size
.
What does this mean?
• irreducible representations: nonabelian analogue of frequencies;
•
-uniform quadrature rule: finite set of sample points that allows for exact
integration, with weights within of uniform;
• B-band-limited function: a function which can be expressed as a linear
combination of “frequencies” from B only;
Point: polynomial-size nearly-uniform quadrature can be achieved generically on
almost any reasonably well-behaved group.
Approximation theory on groups
Results:
Theorem 2. Let (X, w) be an -uniform quadrature rule for End(B) and f, g two B-bandlimited functions. Then
What does this mean?
• End(B) is a larger set than B (it’s the tensor square);
• the quadrature that we achieved in Theorem 1 is actually strong enough to give
something really nice:
• the set of samples is actually a really good geometric proxy for the group; sampling
functions approximately preserves inner products.
These mathematical results prove that certain algorithms (classical or quantum) on
continuous spaces are possible in principle. Designing such algorithms is the next step.
Dissipative preparation of topological states
-- Dissipation is omnipresent in physical systems and provides a
more general description to their evolution.
-- It can in principle be engineered to drive phase transitions,
state engineering or even quantum computation.
Dengis, König,Pastawski
NJP (2014), 16, 1, 013023
König,Pastawski
PRB (2014) 90, 045101
Can we use dissipation to produce topologically entangled states in simple ways?
Do dissipative processes induce a different notion of phases?
Result 1: Encoding into the toric code may be achieved by a
locally generated dissipative process which is almost translation
invariant in a time proportional to the torus perimeter.
Result 2: No locally generated dissipative process may generate
topological ordered states from trivial ones in a time sublinear
in the code distance (assuming non-trivial topology).
Proven using Lieb-obinson bounds and
presence of long range correlations.
Anyons are ``swept’’
in to a fixed sink site.
Tradeoffs for protected gates in quantum codes
-- Guide the search for frugal realizations of faulttolerant quantum computation.
( Where we should not push further and why.)
-- Understand existing results in the context of
general error correcting codes.
-- Classical computers may simulate Clifford group operation and Pauli
measurements efficiently.
-- Magic states + Cliffords enable universal computation
( They can be distilled from non-Clifford gates ).
-- Geometrically local codes are easier to implement experimentally.
-- Geometrically local gates seem a reasonable proxy for fault tolerance.
What can we do with TQFTs?
Topological quantum field theories
What can we do with subsystem codes?
Z5 Z6 Z7 Z1 Z2 Z3 Z4 Z8 Z9
X5 X6 X7 X1 X2 X3 X4 X8 X9
Pastawski, Yoshida
Beverland, Pastawski, Sijher, König, Preskill
Tradeoffs for protected gates in subsystem codes
Pastawski, Yoshida
arxiv:1408.xxxx
Result 1: Stabilizer codes locally defined on a three-dimensional lattice may not
have both of the following properties:
a) An associated Hamiltonian with a macroscopic energy
barrier.
a) A constant depth logical gate outside the Clifford group.
Result 2: The code distance of a D-dimensional stabilizer code
( LD ) with a constant depth implementation of a gate in the
m-th level of the Clifford hierarchy is bounded by O(LD+1-m).
Result 3: Subsystem code families with transversal implementation of gates in the
m-th level of the Clifford have a qubit loss threshold
upper bounded by 1/m.
Result 4: Consider subsystem code families locally
defined in a D-dimensional lattice, having a constant lossthreshold and logarithmically growing code distance.
Then the set of logical unitaries implementable by
constant depth local circuits is included in the D-th level
(Use Hastings 2011 to
of the Clifford hierarchy.
find correctable region)
Locality preserving gates in topological quantum field theories
Beverland, Pastawski, Sijher, König, Preskill
Hypothesis: A deformation lemma for string-like logical
operators is assumed and plays the same role the cleaning
lemma does for stabilizer and subsystem codes.
Result 1 (in progress): For rational abelian anyon models, a
result fully analogous to the one for the toric code is
recovered.
Focus on geometry preservation allows providing a sharper
constraint on the set of allowed ``generalized Cliffords’’.
Result 2 (in progress): Characterization of locality preserving
logical gates on TQFTs supporting non-abelian anyons.
Fibonacci, Ising, … (more restrictive than for abelian case!)
- Local constraints from simple closed loops.
- Global constraints from fusion rules.
- Global constraint from basis change.
Conditional (Quantum) Mutual Information
•
•
Measures correlations between A and B from
the perspective of C:
Nice properties:
–
–
–
•
B
C
Non-negative (strong subadditivity of quantum entropy)
Monotone under local operations
Self-duality (monogamy of entanglement)
Many features known if C is classical but only poorly understand if C is
quantum:
–
•
A
Relation to conditionally independent states (quantum Markov states) or weakly correlated
states?
Important quantity in:
–
–
–
–
Entanglement theory
Quantum coding theory
(Quantum) communication complexity
Condensed matter physics
Renyi Conditional Quantum Mutual Information
•
Berta, Wilde, Seshadreesan (arXiv:1403.6102): introduce continuous Renyi oneparameter family
with
based on αRenyi relative entropy.
•
We show that
keeps the same properties as the original conditional
quantum mutual information:
–
–
–
Non-negative
Monotone under local operations
Self-duality
•
•
So far mostly exploring mathematical properties (heavy matrix analysis).
Possible applications in entanglement theory, quantum coding theory, quantum
communication complexity, condensed matter physics.
•
α-Renyi (relative) entropies are monotone in the parameter α, what about α-Renyi
conditional mutual information?
(important for applications,
proof only for special cases)
Quantum-Proof Pseudorandomness
•
Q: (quantum) observer
The (classical) theory of pseudorandomness:
–
–
–
–
–
Randomness extractors
Randomness condensers
Expander graphs
List-decoding codes
Samplers, Disperser, etc.
Q
Q
N
Ext
•
Example: Randomness extractor.
•
Manifold applications, quantum-proof especially important in:
–
–
–
–
•
•
M
N: input with some entropy
M: perfectly random output
Quantum cryptography
(Classical) post-quantum cryptography
Device independent randomness extraction/amplification
Etc.
Study the power of quantum memory (in otherwise fully classical setup).
Highly non-trivial effects, no generic understanding.
Generic Understanding of Quantum-Proof
•
Spectral randomness extractors are always fully quantum-proof (Berta, Fawzi,
Scholz, Szehr: arXiv:1402.3279).
•
New framework for understanding quantum-proof (Berta, Fawzi, Scholz):
–
–
•
•
Extractor and condenser property is a bounded norm condition between Banach spaces
Quantum-proof extractor and condenser property is a completely bounded norm condition between
operator spaces (non-commutative Banach spaces)
Operator space theory crucial for understanding Bell inequalities.
Many tools from operator space theory to explore, preliminary results (ongoing
work):
–
–
–
Extractors with high entropy input are always approximately quantum-proof
Large separation between bounded and completely bounden norm for condensers (only for the
norms so far)
Condenser property is a special instance of the bipartite densest sub-graph problem (what is
quantum-proof densest sub-graph?)
Future Goals
• Understand mathematical properties of conditional mutual
information: give operational description of quantum states
with small quantum conditional mutual information.
• Explore applications of Renyi conditional mutual information:
– strong converses in quantum coding theory
– lower bounds in quantum communication complexity
– topological order in condensed matter physics
• Pseudorandomness: use operator space theory tools to
generically understand quantum-proof.
• Possible connection to Bell inequalities (two-player games).
• Are specific pseudorandomness constructions quantum-proof
(e.g., Parvaresh-Vardy codes)?
Hypercontractivity of quantum semi-groups
(arXiv: 1403.5224)
Temme,
Pastawski,
Kastoryano
The Question: How fast does a (cptp) semigroup
“smear out” all information about its initial condition and equilibrate?
-- We analyse this by the Hypercontractivity of the semigroup:
where
..
(the weighted 2 --> p-norm measures how spiked the observable f is and
Hypercontractivity indicates that these spikes get smoothed out exponentially fast.)
-- The “differential formulation” of this property is equivalent to a
log-Sobolev inequality
(Here we compare the “Entropy” of an observable with an “Energy like functional” of
the Observable. The constant alpha is called the Log-Sobolev constant.)
The Strategy: We show Hypercontractivity for some fixed p=4 and time
t*. Then we use Riesz-Thorin / Stein-Weiss interpolation Theorems
to obtain bound on the Log.Sobolev constant . This analysis works
well when the generator has “particle like excitations” that are
preserved.
Hypercontractivity of quantum semigroups
(arXiv: 1403.5224)
Temme,
Pastawski,
Kastoryano
So why should we care?:
1) Exponentially improved (over spectral gap) mixing time bounds useful in
dissipative state preparation, analysis of quantum algorithms and
thermalization time estimates.
2) Useful weighted Lp-norm inequalities.
3) Ensures (together with the locality of the generator) stability of the fixed point
under local perturbations. [Cubitt, Lucia, Michalakis, Perez-Garcia]
Results: We prove bounds on the Log-Sobolev constant for
1) Product semigroups with generators:
to estimate 2 ->p norms of product channels:
, which allows
2) Thermalizing Davies generators of:
-- Graph state Hamiltonians. This ensures that the cluster-state of N qubits
can be prepared in time O(polylog(N)), by cooling with T = O(1/log(N)).
-- Free fermion Hamiltonians. (Superconducting wires, etc ...)
Future directions:
-- Rigorous upper and lower bounds to the quantum memory time:
Common figure of merit: “The energy barrier of logical operators”
3D codes with EB: -- Haah code
m = O(log(N))
-- Michnicki’s Welded Toric code m = O(L^(2/3))
But lifetime only scales up to optimal constant value, hence the EB is missing
something.
Open questions:
-- Can we find both upper and lower bounds for the lifetime of quantum
memories ?
-- Can one show that the energy barrier is a necessary criterion for a
thermally stable quantum memory ?
-- Can we derive a better criterion from Poincare type inequalities
for thermalizing Davies semigroups ?
Some results in this direction:
i) O(N) thermalization bound in the Dobrushin - Shlosman regime (i.e. high T)
ii) Lower bounds to the spectral gap of Davies semigroups for stabilizer
codes.
Improved efficiency in variational tensor network algorithms
(G.Evenbly, R. Pfeifer, Phys. Rev. B 89, 245118 (2014))
•
Several theorems relating to the
contraction of generic tensor
networks are presented
These results, when
incorporated into
numerical tensor
network libraries, can
significantly improve
the performance of a
wide variety of
simulation algorithms
e.g. MERA, PEPS +
many more
For an arbitrary
closed network
Theorem 1: (equivalence of environments)
If the environment of one tensor
can be computed with cost 
cost:

If the environment of one tensor
can be computed with cost 
cost:

⇒
⇒
Then the environment of any other tensor
can be computed with the same cost 
cost:
Theorem 2: (multiple environments)

Then set of all environments can
be computed with fixed cost 3
Tensor Network Renormalization Group
(G.Evenbly, work in progress)
•
Efficient coarse-graining algorithms for a large class of D-dimensional tensor networks
•
Basis for efficient numerical algorithms for simulating classical many-body systems
on D-dimensional lattices or quantum systems on (D-1)-dimensional lattices
Initial tensor network
(e.g. representing the
partition function of a
2D classical system)
Insert unitaries and
projectors
Simplify
Coarser tensor network
•
New approach (TNRG) is
effective for critical and gapped
systems (whereas previous
tensor RG schemes could not
properly address critical
systems)
Contract
Research on Quantum Algorithms
at the Institute for Quantum Information and Matter
J. Preskill, L. Schulman, Caltech
preskill@caltech.edu / www.iqim.caltech.edu/
• Progress on last year’s objectives – FY12-13
- Quantum algorithms for simulating particle collisions in fermionic quantum field theories.
- Proposed quantum-resistant cryptosystem based on hardness of solving systems of quadratic equations.
- Quantum circuit obfuscation schemes based on the connections between quantum circuits and braids.
- Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes.
- Efficient algorithm for testing stability of two-dimensional tensor-network states vs. local perturbations.
- Scheme for performing protected quantum gates based on a continuous-variable quantum codes.
- Sufficient condition on noise correlations for scalable quantum computing.
- Near-optimal dynamical decoupling schemes for multi-level quantum systems.
- New class of highly entangled many-body states which can be efficiently simulated.
• Research plan for the next 12 months – FY13-14
-
Quantum algorithms for simulating quantum field theories with gauge fields and massless particles.
Quantum algorithms for simulating thermalization of quantum systems.
Quantum algorithms for interpolating band-limited functions on continuous groups.
Renormalization group analysis of three-dimensional topological quantum codes.
Probability distributions that can be sampled efficiently quantumly but not classically.
Structurally inhomogeneous tensor network states for strongly disordered systems.
• Long term objectives
- Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for
protecting quantum systems from noise.
- Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.
Research on Quantum Algorithms
at the Institute for Quantum Information and Matter
J. Preskill, L. Schulman, Caltech
preskill@caltech.edu / www.iqim.caltech.edu/
• Progress on last year’s objectives – FY13-14
- Quantum algorithms for simulating particle collisions in fermionic quantum field theories.
- Results on optimal contractions of generic tensor networks, with applications to critical systems.
- Optimal algorithms for dissipative preparation of topological code states.
- Characterization of fault-tolerant logical operations in topological subsystem codes.
- Probability distributions that can be sampled efficiently quantumly but not classically.
- Asymptotically uniform, stable, geometry-preserving discretizations of arbitrary compact groups.
- Properties of alpha-Renyi generalizations of quantum conditional mutual information.
- New framework for quantum-proof pseudorandomness based on operator space theory.
- Proof of rapid thermal mixing for graph state Hamiltonians and free fermions.
• Research plan for the next 12 months – FY14-15
-
Show that quantum field theory simulation solves a hard (BQP-complete) problem.
Characterize the logical operators that can be performed using constant-depth unitaries in TQFT.s
Stronger impossibility proofs for quantum obfuscation schemes.
Relate quantum-proofness of randomness condensers to Bell inequality violation.
Show that quantum messages and entanglement increase the power of multi-prover interactive proofs.
Improved rigorous bounds on quantum memory times.
• Long term objectives
- Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for
protecting quantum systems from noise.
- Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.
Some research themes at the IQIM
• Power of quantum computing. Simulating quantum field theories, preparing
thermal states, obfuscating quantum circuits with braids, quantum-resistant
public key based on multivariate quadratic equations, quantum algorithms for
interpolating band-limited functions on continuous groups, probability
distributions that can be sampled efficiently quantumly but not classically.
•Fault-tolerant quantum computing. Magic-state distillation protocol using
triorthogonal quantum codes, RG analysis of self-correcting quantum memory
in 3D, universal topological quantum computing with realistic materials,
protected gates for superconducting qubits, universal dynamical decoupling,
asymmetric Bacon-Shor codes for protection against biased noise.
• Experiment and implementation. Attractive photons in a quantum nonlinear
mediua, Kitaev honeycomb and other exotic spin models with polar molecules,
realizing fractional Chern insulators with dipolar spins.
• Quantum many-body physics. Classifying locally definable quantum
phases, area law and sub-exponential algorithm for 1D systems, fractional
Majorana fermions at the edges of abelian quantum Hall states, class of highly
entangled many-body states which can be efficiently simulated, structurally
inhomogeneous tensor network states for strongly disordered systems.
Obfuscation
G. Alagic,
T. Jeffery,
S. Jordan
Take a circuit C and produce another circuit O(C), so that:
1. functionality is preserved;
2. size is not much bigger (say polynomial);
3. it’s hard to “reverse-engineer” O(C) (at a minimum, O(C) -> C is hard).
Can we have an algorithm that does this for all circuits?
State of affairs in research
– lots of motivation (software/hardware copy protection, homomorphic
encryption, turning private key schemes into public key schemes, etc.)
– known formalizations of (3) are all too hard:
• O(C) no more useful than a black box that performs C?
(impossible, Barak et al ’01)
• O(C1) indistinguishable from O(C2) for equivalent C1, C2?
(collapses PH, Goldwasser Rothblum ’07)
– little is known about quantum obfuscation
• are there classical algorithms for obfuscating quantum circuits?
• are there quantum states that allow us to do obfuscated computation?
Quantum Obfuscation
1.
2.
G. Alagic,
T. Jeffery,
S. Jordan
What if we ask for a slightly weaker condition (3)?
Can we obfuscate quantum circuits?
Results [Alagic Jeffery Jordan ’13]
– efficient classical algorithms for obfuscating both quantum and
classical circuits
– “weaker” condition 3: indistinguishability under a subset of the set of
all circuit relations
Core idea
– if we had an efficient canonical form for circuits (a coNP-hard
problem), we would satisfy Goldwasser-Rothblum trivially
– but topological quantum computation gives us a pretty good mapping
quantum circuits
braids
and braids do have efficient canonical forms!
– in fact, this mapping exists for classical reversible circuits too, if we
use a different representation of the braid group
– If Bob claims to have a quantum computer, Alice can propose that Bob
execute a quantum circuit that obfuscates a classical circuit, where
Alice can easily check the answer.
Approximation theory on groups
G. Alagic,
A. Russell,
L. Schulman
The Discrete Fourier Transform (DFT)
– basis of countless proofs, algorithms, signal processing tasks, etc.
– the fast classical (FFT) algorithms for computing the DFT are very useful in
practice
– their quantum analogues (QFT) are exponentially faster (in a certain
sense) and are a basis for amazing things like Shor’s algorithm
What if the group is continuous instead of finite (say the circle or SU(2))?
– finitely many sums becomes infinitely many integrals.
– two simplifications: only consider band-limited f (doesn’t oscillate too
much), and sample the function at a nicely spaced finite set of points
– for the circle, this boils down to “discretize and use DFT”
Approximation theory on groups
New feature of continuous case:
We can use Fourier inversion to reconstruct the values of the function anywhere
on the group.
Why study the continuous non-abelian case?
– signals in practice might be continuous instead of discrete
– we care about nonabelian spaces (e.g., spherical harmonics, SU(2))
– we need more quantum-algorithmic primitives for exponential speedups
Results [Alagic Russell Schulman 2013]
– a theorem about reconstructing band-limited functions on compact
groups
• setting: any compact group
• input: random samples of a band-limited function f
• output: the list of Fourier coefficients of f
– a number of samples cubic in the band limit is sufficient for a good
estimate
– the reconstruction is inner-product-preserving in the limit
Minimal Updates in Holography (arXiv:1307.0831)
local change in
Hamiltonian
(D+1) – dimensional holographic
description of its ground state
H
IR
H

z
D – dimensional
Hamiltonian
UV
H
Minimal Updates
new ground
state

modified
Hamiltonian
R
H  H  HR
localised change in
holographic description
of ground state


If we use a matrix-product
state description, then a local
change in the Hamiltonian
may require us to modify
tensors far away. With a
holographic description, we
need only modify the tensors
within a causal code of
bounded width.
Evenbly
Tensor network states for disordered systems
Evenbly
Kitaev
An area law and sub-exponential algorithm for 1D systems (Arad, Kitaev,
Landau, Varzirani). Entanglement entropy of gapped 1D system scales
linearly with reciprocol of spectal gap. An algorithm for approximating the
ground state which runs in subexponential time.
Finding the group of units in algebraic number rings of arbitrary degree (in
progress, Eisentraeger, Hallgren, and Kitaev). Toward a uniformly
polynomial algorithm that finds the period of a function on G = Rq for any q.
Classifying locally definable quantum phases of matter (Kitaev). A
definition of quantum phases with short-range entanglement, and a
proposed topological classification of all such phases in any dimension.
Research on Quantum Algorithms
at the Institute for Quantum Information and Matter
J. Preskill, L. Schulman, Caltech
preskill@caltech.edu / www.iqim.caltech.edu/
• Progress on last year’s objectives – FY12-13
- Quantum algorithms for simulating particle collisions in fermionic quantum field theories.
- Proposed quantum-resistant cryptosystem based on hardness of solving systems of quadratic equations.
- Quantum circuit obfuscation schemes based on the connections between quantum circuits and braids.
- Efficient magic-state distillation protocol using a new class of triorthogonal quantum codes.
- Efficient algorithm for testing stability of two-dimensional tensor-network states vs. local perturbations.
- Scheme for performing protected quantum gates based on a continuous-variable quantum codes.
- Sufficient condition on noise correlations for scalable quantum computing.
- Near-optimal dynamical decoupling schemes for multi-level quantum systems.
- New class of highly entangled many-body states which can be efficiently simulated.
• Research plan for the next 12 months – FY13-14
-
Quantum algorithms for simulating quantum field theories with gauge fields and massless particles.
Quantum algorithms for simulating thermalization of quantum systems.
Quantum algorithms for interpolating band-limited functions on continuous groups.
Renormalization group analysis of three-dimensional topological quantum codes.
Probability distributions that can be sampled efficiently quantumly but not classically.
Structurally inhomogeneous tensor network states for strongly disordered systems.
• Long term objectives
- Bring large-scale quantum computers closer to realization by proposing and analyzing new schemes for
protecting quantum systems from noise.
- Conceive, develop, and analyze new applications of quantum computing to physics and mathematics.
Branching MERA
Evenbly, Vidal (1210.1895)
•
•
•
•
Several branches in MERA
Proposed to described highly entangled critical systems.
Cubic code, a gapped spin model, turns out to fit.
Area law holds still, for being in 3D.
Highly entangled quantum circuits (arXiv:1210.1895)
t
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
Block entanglement
entropy scaling
2
2
3
3
T  
T
L
𝑆𝐿 ≈ 𝐿
scales as the bulk
of the block!
IQI Alumni: Postdocs
Eddy Ardonne
Salman Beigi
Sougato Bose
Sergey Bravyi
Darrick Chang
Andrew Childs
Andrew Doherty
Luming Duan
Lukasz Fidkowski
Steve Flammia
Alexei Gorshkov
Sean Hallgren
Patrick Hayden
Liang Jiang
Stephen Jordan
Liang Kong
Robert König
Nordita
IPM
UCL
IBM
ICFO
Waterloo
Sydney
Michigan
Stony Brook
Sydney
NIST
Penn State
McGill
Yale
NIST
New Hamp.
Waterloo
Debbie Leung
Netanel Lindner
Yi-Kai Liu
Ashwin Nayak
Stefano Pironio
David Poulin
Robert Raussendorf
Ben Reichardt
Norbert Schuch
Yaoyun Shi
Kirill Shtengel
Barbara Terhal
Frank Verstraete
Guifre Vidal
Stephanie Wehner
Pawel Wocjan
Shengyu Zhang
Waterloo
Technion
NIST
Waterloo
ULB
Sherbrooke
UBC
USC
Aachen
Michigan
UCR
Aachen
Vienna
Perimeter
Singapore
UCF
Hong Kong
34 former IQI postdocs hold faculty positions (or the equivalent).
13 US, 8 Canada, 7 Europe, 2 Asia, 2 Australia, 2 Middle East
Also: Dave Bacon (Google), Robin Blume-Kohout (Sandia), Sergio Boixo (ISI), Jon Yard (Microsoft)
Coarse graining algorithms for 2D lattice systems
initial
lattice:
unitary
disentanglers
isometries
coarser
lattice:
G. Alagic, S. Jordan
Obfuscation with braids
• to obfuscate a circuit:
1. preserve its functionality;
2. enlarge it by at most a polynomial;
3. make it hard to “reverse-engineer”.
• in practice it seems easy to generate terrible code; in theory,
obfuscation is a notoriously difficult problem.
• choosing the right property 3 is critical: “black-box” obfuscation
is not possible, while “indistinguishability” obfuscation might be;
if two circuits compute the same function, their obfuscated
versions are hard to tell apart.
• we know how to “indistinguishability” obfuscate in computational
models where circuits have a canonical form (ordered binary
decision diagrams).
Quantum circuit obfuscation with braids
Canonical form:
Given a braid B presented as a word in the generators of the braid group,
there is an efficiently computable canonical form. This means that the word
problem in the braid group is easy.
Braid conjugacy:
Given braids A and B expressed as words, and a promise that C exists such
that A = CBC-1, find C. No subexponential algorithm known.
Obfuscation:
quantum circuit
braid
canonical form
obfuscated quantum circuit.
Preserves functionality and efficiency. But does it obfuscate?
Homomorphic encryption:
Encrypt with a random braid B. Circuit is canonical form of BCB-1
| state
B | state
(BCB-1)B | state
BC | state
Testing quantum computers:
Run a classical algorithm through the quantum circuit obfuscator and
challenge the prover to execute the circuit. Not known whether this test can
be foiled by a classical adversary.
G. Alagic, S. Jordan
Outline
1. Basics
2. “Exact” construction
– Distribution can be sampled by quantum computer
– If distribution could be sampled by classical computer,
the PH collapses
3. “Approximate” construction
– Distribution can be sampled by quantum computer
– Even if some “near-by” distribution could be sampled
by classical computer, the PH collapses*
Overview of construction
•
Want to allow for classical sampling of “close-by” distribution
– Formal: Suppose our quantumly sampled distribution is X over {0,1}n, then we would like
the following result:
Given as input ε>0, suppose a classical circuit samples from any
distribution Y, with |X-Y|<ε, in time poly(n,1/ε) then the PH collapses.
•
Problem with prior construction
– Only one exponentially small probability encoded hard information
– If our classical circuit is sampling some “close-by” distribution, it need not include this
information
•
Suppose could “robustly” encode hardness
– Each amplitude encodes Permanent of some matrix
– I could then choose a random y∈{0,1}n and, using approximate counting, estimate:
•
q=Prr[C(r)=y]
– Assuming Permanent is average case hard to estimate, puts P#P⊆Σ2 (but Toda tells us that
PH⊆P#P)
– Again, PH⊆Σ2 (collapse!)
Quantum construction
1.
Define two efficiently computable functions
1.
2.
•
•
2.
g(x,y)=(-1)<x,y>
h:Sn→{0,1}n^2
Takes permutation in Sn to obvious encoding of
permutation as matrix
Note h-1 also efficiently computable and 1-to-1
1
2
Quantum procedure (on n2+log(n!) qubits)
1.
2.
Prepare uniform superposition over log(n!) qubits
Create superposition over h(σ) (by applying h to
first register and h-1 to second)
3.
Pick α at random and apply g(α,h(σ))
4.
Hit with Hadamard on n2 qubits
5.
Measure in standard basis
3
4
This is the permanent of ±1 matrix
encoded by the string α+w
Subtleties
• We get out the permanent of a uniformly chosen ±1 matrix (as
encoded by x+w)
• Suppose we had classical sampler for this distribution
– Using approximate counting, get Σ2 algorithm “computes” the
permanent
1.
2.
Approximate
Average
– *Conjecture 1 & 2 is #P hard
• Approximate by itself known to be hard (1+1/poly mult. approximation)
• Average by itself, can show to be hard!
– For average case reduction to work, need matrix over finite field of “large” size
– For this, replace Hadamard with QFT
– Proof needs some technical work!
• Can generalize Permanent to any VNP complete problem
– Helpful to prove hardness conjectures?
• In fact, even if we had a PH sampler, still collapse hierarchy
Research on Quantum Algorithms
at the Institute for Quantum Information
J. Preskill, A. Kitaev, L. Schulman, Caltech
preskill@caltech.edu / www.iqi.caltech.edu/
Objective
• Improved rigorous estimates of thresholds for
fault-tolerant quantum computation.
• Quantum algorithms beyond the hidden
subgroup paradigm.
• Quantum and classical simulation methods for
quantum many-body systems.
• New approaches to physically robust quantum
computation.
Objective Approach
• Quantum algorithms for simulating local quantum
systems.
• Novel applications of the quantum Fourier
transform and other transforms.
• Customizing quantum fault tolerance for physically
motivated noise models.
• Schemes for physically robust quantum storage and
processing.
• Characterizing Hamiltonian complexity.
• Quantum-resistant classical cryptography.
Efficient quantum algorithm for simulating highenergy particle collisions in strongly-coupled
quantum field theory
Status
• Efficient quantum algorithms for simulating highenergy particle collisions in quantum field theory.
• Proposed trap-door one-way functions based on
tensor problems.
• Enhanced memory time for three-dimensional
quantum memories without string operators.
• Performance analysis for fault-tolerant quantum
computing based on asymmetric Bacon-Shor codes.
• Quantum algorithms for approximating invariants
of triangulated manifolds by tensor contraction.
Open Quantum Systems are stable
• Dissipative quantum processes satisfying rapid mixing are
stable under general perturbations.
• The result is tight: Unstable processes exist if mixing
condition not satisfied.
• First demonstration of stability for Glauber dynamics,
relevant to Markov Chain Monte Carlo methods.
Joint work with T. Cubitt, A. Lucia and D. Perez-Garcia.
Excitations of Gapped Quantum Systems
 Low-energy excitations of translation invariant, gapped Hamiltonians have
elementary-particle-like structure.
 In particular, there exists a family of states that look like localized
excitations on top of vacuum (groundstate), whose overlaps with the lowenergy excitation converge exponentially fast to unity (in the size of the
localized excitation).
 Useful when using DMRG to compute spectral gap and first excited state
in 1D Hamiltonians: Change A-A-A-A-…-A tensor structure of vacuum to
B-A-A-A-…-A and optimize over local B. Then turn into momentum
eigenstate.
Joint work with J. Haegeman, B. Nachtergaele, T. Osborne, N. Schuch
and F. Verstraete.
Stability of Projected Entangled Pair States
 We consider PEPS, believed to provide an efficient description for the low
energy physics arising from local interactions in 2D.
 We provide a checkable criterion for the stability of PEPS under natural
perturbations to the local tensors.
 Criterion is a restriction of Local Topological Quantum Order (LTQO), the
principal condition for the stability of frustration-free, gapped Hamiltonians.
 LTQO is shown to be stable under perturbations, implying it is a generic
property of gapped phases of matter.
Joint work with I. Cirac, D. Perez-Garcia and N. Schuch.
Kitaev
Locally definable phases
Area law.
G. Alagic, S. Jordan
Obfuscation with braids
• to obfuscate a circuit:
1. preserve its functionality;
2. enlarge it by at most a polynomial;
3. make it hard to “reverse-engineer”.
• in practice it seems easy to generate terrible code; in theory,
obfuscation is a notoriously difficult problem.
• choosing the right property 3 is critical: “black-box” obfuscation
is not possible, while “indistinguishability” obfuscation might be;
if two circuits compute the same function, their obfuscated
versions are hard to tell apart.
• we know how to “indistinguishability” obfuscate in computational
models where circuits have a canonical form (ordered binary
decision diagrams).
Quantum circuit obfuscation with braids
Canonical form:
Given a braid B presented as a word in the generators of the braid group,
there is an efficiently computable canonical form. This means that the word
problem in the braid group is easy.
Braid conjugacy:
Given braids A and B expressed as words, and a promise that C exists such
that A = CBC-1, find C. No subexponential algorithm known.
Obfuscation:
quantum circuit
braid
canonical form
obfuscated quantum circuit.
Preserves functionality and efficiency. But does it obfuscate?
Homomorphic encryption:
Encrypt with a random braid B. Circuit is canonical form of BCB-1
| state
B | state
(BCB-1)B | state
BC | state
Testing quantum computers:
Run a classical algorithm through the quantum circuit obfuscator and
challenge the prover to execute the circuit. Not known whether this test can
be foiled by a classical adversary.
G. Alagic, S. Jordan
Quantum algorithms for quantum field theories
-- Feynman diagrams have limited precision,
particularly at strong coupling.
-- Classical lattice methods can compute
static properties, but cannot simulate
dynamics
A quantum computer can simulate particle collisions, even at high energy
and strong coupling, using resources (number of qubits and gates) scaling
polynomially with precision, energy, and number of particles.
-- Estimate errors due to regulating (spatial lattice and approximating
continuous variable fields by qubits).
-- Efficient procedure for preparing (strongly-coupled) vacuum and initial
wave packet states, simulating time evolution, measuring final state.
Does the quantum circuit model capture the
computational power of Nature?
What about quantum gravity?
Jordan, Lee, Preskill, SCIENCE, 336: 1130, 1 JUNE 2012
Simulating quantum field theory
Input: a list of incoming particle momenta (particles are actually wave
packets with some momentum spread).
Output: a list of outgoing particle momenta.
Goal is to sample accurately from the distribution of final state particles that
would be produced in a high energy collision in a (strongly coupled) field
theory.
Consider a self-coupled scalar field in d = 1, 2, 3, spatial dimensions.
Procedure:
(1) Prepare free field vacuum.
(2) Prepare free field wavepackets.
(3) Adiabatically turn on the coupling constant (t).
(4) Evolve for time T using interacting Hamiltonian H.
(5) Adiabatically turn off coupling
(6) Measure field modes of free theory.
Need to discretize the problem, and keep track of resulting errors.
Jordan, Lee, Preskill
Simulating quantum field theory
Define Hamiltonian on spatial lattice:
1
H (a )   a d  2  ( ) 2  m 2 2   4 
2 x
Successively finer lattice approach continuum, m2 and are functions of a,
and additional terms in the Hamiltonian apprear as a changes.
….
Finer lattice means higher accuracy but more qubits.
Physical mass of particles assumed positive.
Field at each lattice space is real variable, digitized using nb qubits, where
nb scales logarithmically with energy and accuracy.
Simulating quantum field theory
Jordan, Lee, Preskill
Free theory is Gaussian – there is efficient algorithm for preparing the
Gaussian vacuum state – then simulate interaction of field mode with a two
level “atom”:
†
H g ,e  a | g e |  a | e g |
Turning on the coupling: Trotter approximation of adiabatic evolution, while
wave packet is “frozen” to prevent spreading and propagating.
Choose turn-on time to suppress unwanted diabatic processes.
Particle creation: energy gap ~ mphysical
Particle splitting: energy gap ~ (mphysical)2 / momentum
Measurement: after turning off coupling, use phase estimation to determine
particle number in each mode.
Discretization error scales like a2, where a is lattice spacing. Number of
lattice sites ~ a-d, and # gates in Gaussian state preparation ~ ( -d/2)2.376
Diabatic error determines evolution time as coupling turns on. Number of
gates ~ E 2( d 1) / m8phys where E is total energy.
Of order 104 qubits for 1% accuracy in one spatial dimension.
Simulating quantum field theory
Next steps:
Include fermions.
Massless particles (infrared safe observables).
Gauge fields (start with strong coupling limit).
Simulate standard model of particle physics in BQP.
Quantum gravity?
Noise correlations and scalability
In general, the noise Hamiltonian may contain terms acting on m system
qubits, for m = 1, 2, 3, ….
H System  Bath   H i(1)   H ij(2)   H ijk(3) 
 ij 
i
 ijk 
Quantum computing is provably scalable if ε ≤ ε0
  (9.44)  max m m1/ m
m  max j
1
over all times
and qubits

and
‖ H ( m ) j1 j2 j3 jm ‖ t0
j2 , j3 ,, jm
interactions fall
off with distance
10-4, where
[ t0 is the maximal duration
of any quantum gate. ]
term that acts collectively on m system
qubits should be exponentially small in m.
Currently known proofs of the threshold theorem require the noise to be
“quasi-local” in the sense that the m-qubit noise term in the Hamiltonian
decays exponentially with m. Can experiments verify this scaling?
J. Preskill