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Quantum Frontiers in Computer Technology
Jacob Biamonte
www.QuamPlexity.org
Talk given at Skolkovo Institute of Science and Technology
Our approach to utilize & understand quantum effects
physics of informationmathematical
network theory
quantum algorithms
tensor networks
local Hamiltonian complexity
quantum dynamical systems
networkinformation theory
condensed matter
quantumcomplexity science
QuamPlexity.org 1
The thesis of today’s talk
Our work advocates that ground states of realizable systems can be
harnessed as a powerful and naturally prevalent resource to enhance
computation using quantum effects.
Moreover, the method of combinatorial optimization using Hamiltonian
ground states is capable of functioning without quantum effects and can
be improved incrementally with their inclusion.
We argue that while combinatorial optimization provides a tangible path
to further precision engineering and quantum control, the development of
a universal ground state quantum computer will be essential for
significant computational gains.
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Table of contents
1. Quantum Annealing
2. Universal Ground States for Quantum Computation
3. Quantum Computation of Molecular Energy Simulation
4. Simulation by Tensor Contraction
5. Conclusion
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Quantum Annealing
What is quantum annealing?
• Quantum annealing is a quantum enhanced method to solve
combinatorial optimization problems1
• Many problems in machine learning can be formulated as
combinatorial optimization including e.g. sampling Boltzmann-Gibbs
distributions
• D-Wave Systems sells hardware to realize quantum annealing with
O(103) spins
• Customers include Google, NASA, Lockheed-Martin and Los Alamos
National Laboratory
1Brooke, Bitko, Rosenbaum and Aeppli, Science 284:779 (1999)
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How does quantum annealing work?
• Minimizes a cost function by controlling quantum fluctuations
(tunneling)
• To perform quantum annealing, one maps the cost function (called
pseudo Boolean form) to a tunable quadratic Ising model
(i.e. magnetism)
H =∑
JijZiZ j +
∑hkZ
k (1)
• The Hamiltonian (energy function) of the tunable Ising model is
chosen by assignment of Jij , hk such that its lowest-energy state
(ground state) represents the unknown solution to the problem
instance
• Non-commuting local X terms are added during the computation to
induce quantum tunneling transitions
H ′ =∑
JijZiZ j +
∑hkZ
k +∑
∆lXl (2)
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How do you program a quantum annealer?
• The traditional approach was to embed graph problems.
• In practice, this was cumbersome and sometimes existing methods
were not ideal for specific hardware constraints
• We therefore developed methods to represent logic gates in ground
states, and hence could map ground state energy problems to circuit
SAT directly2,3
• This had several advantages and provided penalty functions
representing central building blocks in wide use today.
2Whitfield, Faccin and Biamonte, EPL 99:57004 (2012)3Biamonte, Physical Review A 77:052331 (2008)
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The team
• Non-perturbative k-body to two-body commuting conversion
Hamiltonians and embedding problem instances into Ising
spins
Physical Review A 77:052331 (2008)
• Ground State Spin Logic
Whitfield, Faccin and Biamonte
EPL (Europhysics Letters) 99:57004 (2012)
• Hamiltonian Gadgets with Reduced Resource Requirements
with Cao and Kais
Physical Review A 91:012315 (2015)
Yudong Cao Mauro Faccin James WhitfieldQuamPlexity.org 7
Ground state AND penalty
x1 x2 z? z??= x1 ∧ x2 H∧(x1, x2, z?)
0 0 0 〈000|H∧ |000〉 = 0 0
0 0 1 〈001|H∧ |001〉 ≥ δ 3δ
0 1 0 〈010|H∧ |010〉 = 0 0
0 1 1 〈011|H∧ |011〉 ≥ δ δ
1 0 0 〈100|H∧ |100〉 = 0 0
1 0 1 〈101|H∧ |101〉 ≥ δ δ
1 1 0 〈110|H∧ |110〉 ≥ δ δ
1 1 1 〈111|H∧ |111〉 = 0 0
(left) assignments of the variables x1, x2 and z?. (center) variable
assignments that receive an energy penalty ≥ δ. (right) truth table for
H∧(x1, x2, z?) = 3z? + x1 ∧ x2 − 2z? ∧ x1 − 2z? ∧ x2, with null space
L ∈ span{|x1x2〉 |z?〉 |z? = x1 ∧ x2,∀x1, x2 ∈ {0, 1}}.
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ZZZ penalty4,5
• A three-parameter family of Hamiltonians encoding NAND is
Hx∨y (Z 1,Z 2,Z 3) = (c1Z1 + c2Z
2)(1 + Z 3) (3)
+(c1 + c2)Z 3 + c12
∑i<j
Z iZ j
with c1, c2, c12 > 0. The parameter freedom reduces experimental
constraints
• We arrive at the following three-parameter family that preserves the
ground state subspace of XOR (ZZZ)
Hzzz = Hx∧y (Z 1,Z 2,Z 4)− Z 3 (4)
+Z 1Z 3 + Z 2Z 3 + 2Z 3Z 4
• The coefficients, c1, c2, c12, must be greater than 1/2 instead of
strictly positive4Whitfield, Faccin and Biamonte, EPL 99:57004 (2012)5Biamonte, Physical Review A 77:052331 (2008)
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Implications
• Using the penalty for ZZZ, one can reduce any 3-body Ising
Hamiltonian to a 2-body one
• Can be chained to transform k-body terms to 2-body ones, with the
cost of k − 1 ancilla spins6,7
• Using the penalty functions for e.g. NAND, one establishes an
alternative means to prove that Ising models with a suitable range of
couplings are NP-hard where a suitable decision problem exists,
establishing NP-completeness
• The physical Turing principle states that a universal computing
device can simulate every physical process (note that not all natural
systems settle to their ground states in accessible time—e.g. glassy
systems)
6Whitfield, Faccin and Biamonte, EPL 99:57004 (2012)7Biamonte, Physical Review A 77:052331 (2008)
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Limitations of quantum enhanced optimization
• Connectivity—how do we embed problem instances into actual
hardware?
• How hard is it to catch up with existing technology?
• Performance unclear—gate model gives ∼ O(√N) for search in the
oracle model
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Universal Ground States for
Quantum Computation
It’s hard to prove what we can’t do
• In the way that we proved that Ising systems are classically universal
(i.e. can embed and hence simulate classical circuits), we want to
study non-diagonal Hamiltonians which are universal for quantum
computing
• In the case of non-diagonal Hamiltonians, we can either say (i) that
they are as powerful as quantum computers, or (ii) that we can
simulate them using polynomial resources classically
• It is widely believed, based on empirical evidence and analytical toy
models (e.g. the oracle model) that classical computers can not
efficiently simulate quantum ones
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Stochastic Hamiltonians
Stoquastic Hamiltonians, those for which all off-diagonal matrix elements
in the standard basis are real and non-positive (eqv. non-negative), are
common in the physical world
• Theory merges elements of quantum mechanics and the classical
theory of stochastic matrices
• For non-technical purposes ‘stoquastic’ is equivalent to avoiding the
sign problem
Recall that the physical Turing principle states that a universal
computing device can simulate every physical process.
We have studied in detail such processes, in both the quantum and
stochastic settings.
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Dynamics with Stochastic Hamiltonians
A : Adjacency matrix
(non-diagonal, unlike Ising
model)
D : Matrix with node degrees on
the diagonal
L : Laplacian matrix
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Stochastic Process
• start from a node
• choose a neighbor
• move to it
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Stochastic Process
• start from a node
• choose a neighbor
• move to it
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Stochastic Process
• start from a node
• choose a neighbor
• move to it
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Stochastic Process
• start from a node
• choose a neighbor
• move to it
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Degree and Stochastic Processes
Stochastic Generator:
HC = LD−1
Probability distribution at time t:
PC (t) = eHC tPC (0)
The eigenvector with zero
eigenvalue is:
φ0 =
d1
d2
...dn
Linear correlation between degree
and the stochastic steady state
probability distribution
Degree
(PC )i
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Quantum Comparison
Quantum
Sto
chas
tic
Spectral S
imil
ar
Quantum Generator
HQ = D−12LD− 1
2
with the same spectrum of LD−1.
The eigenvector corresponding to
the zero eigenvalue is:
φ0 =
√d1√d2
...√dn
• initial state
• long time average
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Comparison of stochastic vs quantum processes
• Quantum Techniques for Stochastic Mechanics
John C. Baez and Jacob Biamonte
(to appear in World Scientific) 274 pages arXiv:1209.3632 (2017)
• Complex Networks: from Classical to Quantum
Jacob Biamonte, Mauro Faccin and Manlio De Dominico
in review (2017) arXiv:1702.08459
• Spectral Entropies as Information-Theoretic Tools for Complex
Network Comparison
Manlio De Domenico and Jacob Biamonte
Physical Review X 6:041062 (2016)
• Community Detection in Quantum Complex Networks
Faccin, Migdal, Johnson, Bergholm and Biamonte
Physical Review X 4:041012 (2014)
• Degree Distribution in Quantum Walks on Complex Networks
Faccin, Johnson, Kais, Migdal and Biamonte
Physical Review X 3:041007 (2013)QuamPlexity.org 18
Comparison of stochastic vs quantum processes
John Baez Ville Bergholm Manlio De Dominico
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Annealing vs adiabatic quantum computing
• The adiabatic theorem can be used to devise quantum algorithms
that make use of ground states8
• For example, consider monotonic s ∈ [0, 1],
H(s) = (1− s)Hi + s · Hf (5)
• The initial state for combinatorial optimization is a summation over
basis states
2−n/2∑
x∈{0,1}n|x〉 (6)
which is the unique ground state of Hi = −∑
k Xk
• The evolution depends on the spectrum of H(s) which depends on
the path taken; for universal adiabatic quantum computation, we
have to prove that an adiabatic theorem is satisfied8Farhi, Goldstone, Gutmann, Lapan, Lundgren, and Preda, Science 292:472 (2001)
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What Hamiltonian has a universal ground-state for quantum
computing?
What are the minimal physical resources required for universal quantum
computation?
Realizable Hamiltonians for Universal Adiabatic Quantum
Computers
Biamonte and Love
Physical Review A 78, 012352 (2008)
Theorem 1. The ground state energy problem for the ZX Hamiltonian is
QMA-complete
HZX =∑
hiZi +
∑∆jX
j +∑
JklZkZ l +
∑KmnX
mX n (7)
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Proof sketch
The procedure to prove universality embeds the history of a general
quantum circuit into a local-Hamiltonian.
The method further requires the use of perturbation theory to simulate
3-body (and higher) interactions using two-body ones which is done by
coupling to ancillary qubits.
The perturbation is taken with respect to the ancillary qubits—to which
a large energy gap must be applied. For an error ε the gap can be shown
to scale as some inverse polynomial O(ε−k).
We improved the gaps of all the known gadgets (including those
introduced in my papers)9
9with Yudong Cao et al., Physical Review A 91:012315 (2015)
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Universal does not imply practical
The entire procedure to establish universality requires significant
overhead and I personally argue does not appear to represent a practical
means towards universal ground state quantum computation.
Developing alternative methods to program such devices represents a
problem of both practical and theoretical importance.
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An application
In his famous 1981 talk, Feynman proposed that when simulating
quantum phenomena, a universal quantum simulator would not
experience an exponential slowdown.
Adiabatic Quantum Simulators
Biamonte, Bergholm, Whitfield, Fitzsimons and Guzik
AIP Advances 1, 022126 (2011)
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Adiabatic quantum simulators
Our method works by preparing a initially non-interacting probe P and
system of interest S into their ground states.
The measurement procedure begins by bringing S and P adiabatically
into interaction.
The system and probe Hamiltonians increases in locality by one.
A measurement procedure similar to Ramsey spectroscopy is then used to
recover the lowest eigenvalue.
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Adiabatic quantum simulator readout
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t ω0
P0
Ramsey measurement of the probe
simulation
fit
Measurement procedure under simulated Markovian noise. The
continuous curve represents an ideal model, the circles are averaged
measurement results and the dotted line a least squares fit to them.
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Quantum Computation of
Molecular Energy Simulation
Why electronic structure?
The lack of computationally efficient methods for the accurate simulation
of quantum systems on classical computers presents a grand challenge.
Note that several faculty members at Skoltech actively research and have
pushed forward the state of the art in such simulations.
Recall that the physical Turing principle states that a universal
computing device can simulate every physical process.
Conjecture. Molecules naturally reside in their low-energy configuration,
so a quantum computer might probe this low-energy subspace efficiently.
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Some Results
Quantum Simulation of Helium Hydride Cation in a Solid-State
Spin Register
with Ya Wang et al.
ACS Nano 9:7769 (2015)
High fidelity spin entanglement using optimal control
with Dolde et al.
Nature Communications 5:3371 (2014)
Simulation of Electronic Structure Hamiltonians using Quantum
Computers
with Whitfield and Guzik
Molecular Physics 109:735 (2011)
Towards Quantum Chemistry on a Quantum Computer
with Lanyon et al.
Nature Chemistry 2:106 (2009)
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Practicality of Quantum Computing Electronic Structure?
The overheads to execute these algorithms requires O(102) qubits and
high connectivity.
It is not clear how error correction will be incorporated into these systems
and it is further not clear how useful these devices will be—if at
all—without error correction.
It is generally not known what the impact of noise will be on
non-stoquastic quantum annealing.
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A Roadmap to Quantum Compute Electronic Structure?
Despite some difficulties, using quantum computers for electronic
structure calculations has realistic promise and the first steps towards this
goal will be to construct quantum annealers.
This is of course already being worked on as part of a vast global effort.
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Simulation by Tensor Contraction
Tensor contractions outside of quantum computing
Tensor network methods are taking a central role in modern quantum
physics and computer science and are now being used to enhance
machine learning.
Tensor Networks for Dimensionality Reduction and Large-scale
Optimization: Part 1 Low-Rank Tensor Decompositions
Cichocki, Lee, Oseledets, Phan, Zhao and Mandic
Foundations and Trends in Machine Learning 9:4-5 249 (2016)
Tensor Networks for Dimensionality Reduction and Large-scale
Optimization: Part 2 Applications and Future Perspectives
Cichocki, Phan, Zhao, Lee, Oseledets, Sugiyama and Mandic
Foundations and Trends in Machine Learning 9:6 431 (2017)
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Tensor networks in quantum physics
These methods can provide an efficient approximation to certain classes
of quantum states, and the associated graphical language makes it easy
to describe and pictorially reason about quantum circuits, channels,
protocols, open systems and more.
Quantum Tensor Networks in a Nutshell
Biamonte and Bergholm
to appear in Contemporary Physics (2017)
Tensor Network Contractions for #SAT
Biamonte, Turner and Morton
Journal of Statistical Physics 160, 1389 (2015)
Tensor Network Methods for Invariant Theory
Biamonte, Bergholm, and Lanzagorta
Journal of Physics A: Mathematical and Theoretical 46, 475301 (2013)
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Tensor contraction and quantum computing
A quantum computer is basically a linear algebra accelerator capable of
contracting tensors that seem out of reach by classical devices.
Are these contractions actually out of reach?
Any serious effort to build a quantum computer that aims to outperform
the best classical devices, must overcome the best numerical algorithms.
In that regard, tensor network simulations can inform the design of a
future quantum computer by determining classically difficult problems.
QuamPlexity.org 33
Conclusion
The ground-state golden thread
physics of informationmathematical
network theory
quantum algorithms
tensor networks
local Hamiltonian complexity
quantum dynamical systems
networkinformation theory
condensed matter
quantumcomplexity science
QuamPlexity.org 34
Summary
• A ground state of a physical system can embed and simulate digital
switching circuits or—depending on the Hamiltonian—embed
quantum circuits
• No practical method exists to program a universal ground state
quantum computer; however methods exist to use ground states to
simulate quantum systems
• The simulation of electronic structure seems to be the strongest
potential future application of quantum simulators
• Tensor network algorithms compress the data required to simulate
quantum physics. A practical quantum computing demonstration
must outperform the best tensor network simulation algorithms
• Tensor network algorithms are ripe to be employed in areas outside of
quantum computing and condensed matter physics, with application
potential in image/data compression and machine learning
QuamPlexity.org 35
Sponsors
QuamPlexity.org 36
References I
Embodiment of a universal adiabatic quantum computer, 2007.
U.S. Patent 60/910, 445 (2007).
V. Bergholm and J. Biamonte.
Categorical quantum circuits.
Journal of Physics A: Mathematical and Theoretical, 44(24):245304,
2011.
J. Biamonte.
Nonperturbative k-body to two-body commuting conversion
hamiltonians and embedding problem instances into ising
spins∗∗.
Physical Review A, 77(5):052331, 2008.
QuamPlexity.org
References II
J. Biamonte.
Charged string tensor networks∗∗.
Proceedings of the National Academy of Sciences,
114(10):2447–2449, 2017.
J. Biamonte and V. Bergholm.
Tensor networks in a nutshell.
to appear.
J. Biamonte, V. Bergholm, and M. Lanzagorta.
Tensor network methods for invariant theory.
Journal of Physics A: Mathematical and Theoretical, 46(47):475301,
2013.
QuamPlexity.org
References III
J. Biamonte, V. Bergholm, J. Whitfield, J. Fitzsimons, and
A. Aspuru-Guzik.
Adiabatic quantum simulators∗∗.
AIP Advances, 1(2):022126, 2011.
J. Biamonte, S. Clark, and D. Jaksch.
Categorical tensor network states.
AIP Advances, 1(4):042172, 2011.
J. Biamonte, M. Faccin, and M. De Domenico.
Complex Networks: from Classical to Quantum.
arXiv:1702.08459—in review, Feb. 2017.
J. Biamonte and P. Love.
Realizable hamiltonians for universal adiabatic quantum
computers∗∗.
Physical Review A, 78(1):012352, 2008.
QuamPlexity.org
References IV
J. Biamonte, J. Morton, and J. Turner.
Tensor network contractions for #SAT.
Journal of Statistical Physics, 160(5):1389–1404, 2015.
Y. Cao, R. Babbush, J. Biamonte, and S. Kais.
Hamiltonian gadgets with reduced resource requirements.
Physical Review A, 91(1):012315, 2015.
M. De Domenico and J. Biamonte.
Spectral entropies as information-theoretic tools for complex
network comparison∗∗.
Physical Review X, 6(4):041062, 2016.
S. Denny, J. Biamonte, D. Jaksch, and S. Clark.
Algebraically contractible topological tensor network states.
Journal of Physics A: Mathematical and Theoretical, 45(1):015309,
2012.
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References V
F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov,
S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann,
T. Schulte-Herbruggen, and J. Biamonte.
High-fidelity spin entanglement using optimal control∗∗.
Nature Communications, 5, 2014.
M. Faccin, T. Johnson, J. Biamonte, S. Kais, and P. Migda l.
Degree distribution in quantum walks on complex networks∗∗.
Physical Review X, 3(4):041007, 2013.
M. Faccin, P. Migda l, T. H. Johnson, V. Bergholm, and
J. Biamonte.
Community detection in quantum complex networks∗∗.
Physical Review X, 4(4):041012, 2014.
QuamPlexity.org
References VI
R. Harris, A. Berkley, M. Johnson, P. Bunyk, S. Govorkov, M. Thom,
S. Uchaikin, A. Wilson, J. Chung, E. Holtham, and J. Biamonte.
Sign-and magnitude-tunable coupler for superconducting flux
qubits∗∗.
Physical Review Letters, 98(17):177001, 2007.
T. Johnson, J. Biamonte, S. Clark, and D. Jaksch.
Solving search problems by strongly simulating quantum
circuits.
Scientific Reports, 3, 2013.
B. Lanyon, J. Whitfield, G. Gillett, M. Goggin, M. Almeida,
I. Kassal, J. Biamonte, M. Mohseni, B. Powell, M. Barbieri, et al.
Towards quantum chemistry on a quantum computer∗∗.
Nature Chemistry, 2(2):106–111, 2010.
QuamPlexity.org
References VII
D. Lu, J. Biamonte, J. Li, H. Li, T. H. Johnson, V. Bergholm,
M. Faccin, Z. Zimboras, R. Laflamme, J. Baugh, et al.
Chiral quantum walks.
Physical Review A, 93(4):042302, 2016.
S. Meznaric and J. Biamonte.
Tensor networks for entanglement evolution.
Advances in Chemical Physics, 154:561–574, 2012.
J. Morton and J. Biamonte.
Undecidability in tensor network states.
Physical Review A, 86(3):030301, 2012.
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References VIII
Y. Wang, F. Dolde, J. Biamonte, R. Babbush, V. Bergholm, S. Yang,
I. Jakobi, P. Neumann, A. Aspuru-Guzik, J. D. Whitfield, et al.
Quantum simulation of helium hydride cation in a solid-state
spin register∗∗.
ACS Nano, 9(8):7769–7774, 2015.
J. Whitfield, J. Biamonte, and A. Aspuru-Guzik.
Simulation of electronic structure hamiltonians using quantum
computers∗∗.
Molecular Physics, 109(5):735–750, 2011.
J. Whitfield, M. Faccin, and J. Biamonte.
Ground-state spin logic.
EPL (Europhysics Letters), 99(5):57004, 2012.
QuamPlexity.org
References IX
C. Wood, J. Biamonte, and D. Cory.
Tensor networks and graphical calculus for open quantum
systems.
Quantum Information & Computation, 15(9&10):759–811, 2015.
Z. Zimboras, M. Faccin, Z. Kadar, J. D. Whitfield, B. Lanyon, and
J. Biamonte.
Quantum transport enhancement by time-reversal symmetry
breaking∗∗.
Scientific Reports, 3, 2013.
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