Download High Performance Quantum Computing

High Performance
Quantum Computing
Matthias Troyer
Matthias Troyer
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Beyond exascale: quantum devices
Quantum random numbers
perfect randomness
Quantum encryption
secure communication
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Quantum computers?
solve quantum models (R. Feynman)
factor integers (P. Shor)
break RSA encryption
solve linear equations (A. Harrow et al)
Analog quantum simulators
solve quantum models
...
Quantum annealer
solve hard optimization problems?
Matthias Troyer
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Quantum random numbers
Quantum mechanics can give true and perfect random numbers
0
0
photo detectors
1
1. Photon source emits a photon
2. Photon hits semi-transparent mirror
3. Photon follows both paths
semi-transparent mirror
4. The photo detectors see the photon only
in one place: random selection
photon source
5. Record one random bit
Matthias Troyer
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Quantum cryptography
Exchange a secret key to encrypt a message using entangled photons
No eavesdropper can listen without being detected
except for the quantum hacker Vadim Makarov (Waterloo)
Quantum hacking lab
Matthias Troyer
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D-Wave Two quantum annealer
Solves NP-complete spin glass problem with up to 512 variables
H = ∑ J ij si s j + ∑ hi si + const.
ij
i
with
si = ±1
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1
Can be built with imperfect qubits
Suffers form calibration problems like other analog devices
Unknown if this technology can ever scale better than classical devices
Matthias Troyer
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What about quantum computers ?
?
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Matthias Troyer
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Classical versus quantum bits
▪ Classical bits can only be either 0 or 1
▪ Quantum bits can have both values at once, in arbitrary superpositions
ψ =α 0 +β 1
▪
2
need two complex or three real numbers to describe the state
⎛ α ⎞
ψ =⎜
⎟
β
⎝
⎠
▪
0
α + β =1
2
⎛θ ⎞
α = cos ⎜ ⎟
⎝ 2⎠
⎛θ ⎞
β = e sin ⎜ ⎟
⎝ 2⎠
iφ
1
but when measuring (looking) I only ever get 1 bit
0 with probability α
2
1 with probability β
2
Matthias Troyer
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Information content of a quantum register
▪ The state of an N qubit register
▪ needs 2N complex numbers to be represented classically
▪ but when measured only gives N bit of information
▪ Advantages and limitations of quantum computers
▪ Exponential intrinsic parallelism: operate on 2N inputs at once
▪ But very limited readout of only N bits
▪ No-cloning theorem: a quantum register cannot be copied
Try to build a
cloner for 0 and 1
C0 → 0 0
C1 →1 1
Linearity of quantum mechanics means it
will not clone an arbitrary state
C (α 0 + β 1 ) → α 0 0 + β 1 1
≠ (α 0 + β 1
)(α 0
+β 1
)
Matthias Troyer
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necessary. In the following I will discuss a set!of typically
used one an
"
and will then discuss which ones are strictly necessary.
A set of common
two-qubit
gates are
controlled
gates,
consisting
0 of
1 a control qubit an
and will
then discuss
which
ones are
strictly
necessary.
(NOT) version CU
X of a single qubit 1gate
a target qubit.Pauli-X
The controlled
0 "U (any of the li
!
Single
qubit qubit
gatesoperation U on the target0qubit
above) performs
the single
−i only if the contr
Single
qubit
gates
Pauli-Y
!
"
Y
qubit is set to 1.
i
0
! 0 1 ""
!
Quantum gates
Pauli-X
The quantum
circuit(NOT)
for such a gate is:X
01 010
1
Pauli-X
Pauli-Z(NOT)
! 1 0 "
XZ•
00 −1
−i ""
!
!
Pauli-Y
01i −i
1
Y
the
NAND
gate
is
universal
▪ In classical computing
1
0
√
Pauli-Y
Hadamard gate
! i 0 "
Y UH
2
11 −1
0 ""
!
!
Pauli-Z
10 1−1
00
Z gates
we need three
▪ In quantum computing
Pauli-Z
Phase gate
! is "
ZS , |01⟩ , |10⟩ , |11⟩}
Its matrix representation in a basis {|00⟩
i ""
01 0−1
1
!
!
1
Hadamard gate ⎛
⎞ √
H
10
11 2 111 −1
1
√
1
0
0
0
T gate or gate
π/8 gate
Hadamard
!
"
1
→
0 −1)
iπ/4
(
0
→
0
+
1
)
T
2 ( 01 e−1
▪ Hadamard gate (flips x and z) ⎜ 0 1 H
2
⎟
2 ! 1 0"
!
"
0
0
⎜
⎟
Phase gate
. e−iθ/2 10 00i
(8.5
S
⎝
⎠
0
0
Rz(θ)gate
gate
!
Phase
Rz(θ)
iθ/2 "
S U
0
e
0
1
0 i " iπ /4
0
0
!
T gate or π/8 gate
iπ/4
α 0 + β101 e→
α 0 +e β 1
T
0
Ato
few
remarks
be useful. The
gate is the "
quantum analog
! X−iθ/2
phase
state)
▪ T gate (applies
Timportant
gate
or1π/8
gatemay
The most
two-qubit
gate isTthe controlled-NOT-gate
iπ/4 (CNOT ), which
0 eidentity
eto the
0 " and is essential
gate.
The
Hadamard
gate
(H)
squares
!
Rz(θ)
gate
the same as a controlled-X gate. It is typically
−iθ/2
Rz(θ) drawneas
iθ/2
0 aligned
e0
rotation
around
the
y
axis,
rotating
a
state
with z to x. T
Rz(θ) gate
Rz(θ)
iθ/2
•
0
e
called
π/8remarks
gate since
can
– up to
anX
global
– be wo
A few
mayitbe
useful.
The
gate
quantum
x irrelevant
y →isxthe
x⊕
y phaseanalog
flips)
▪ CNOT gate (conditionally
" is essential
!gate
gate.
The
Hadamard
(H) squares
the
identity
and
A few
remarks
may gate
be useful.
The Xto
is
the quantum
analog o
−iπ/8
e identity
0 with
iπ/8
rotation
the gate
y axis,
a
state
aligned
to x. T
gate.
The around
Hadamard
(H)rotating
squares
to
the
and. isz essential
T =e
iπ/8
0aligned
e with
The matrix
representation
is y itaxis,
called
π/8
gate the
since
can rotating
– up to an
irrelevant
global
phase
be w
rotation
around
a state
z to– x.
T
This
choice
is
not
unique
and
having
more
gates
can
make
a
▪
⎛
⎞an irrelevant global phase – be w
called
π/8
gate
since
it
can
–
up
to
" spin space.
! −iπ/8
1
0
0
0
The
Rz
gate
performs
a
rotation
around
the
z
axis
in
device more efficient
e
0
iπ/8
⎜
⎟
"Ry
!
. gates.
0 are
1 performed
0T =
0 e⎟
around the x and y ⎜
axis
by
the
Rx
and
iπ/8
−iπ/8
.
0
e
e
0
⎝ 0 0 T0 =1e⎠
iπ/8
| 10
iπ/8 | .
0
e
0 1 0 around 78
The Rz gate performs 0a rotation
the z axis in spin space.
Matthias Troyer
sch algorithm and its generalization, the Deutsch-Jozsa algorithm are the
uantum algorithms that show an advantage over classical algorithms, even
e problem they solve is somehow artificial. You are given a binary function
n values are either 0 or 1) and know that either the function is constant, or
ced, i.e. it is 0 for exactly half the inputs and 1 for the other half. The
nd Deutsch-Jozsa algorithm can decide between the two cases with exactly
on call.3
The Deutsch algorithm: simplest quantum speedup
▪ Check whether a binary function is constant or not
tsch algorithm
sch algorithm asks the question
to decide
whether a binary function of one
f :{0,1}
→ {0,1}
able f : {0, 1} → {0, 1} is balanced or constant. Classically one has to
make▪twoClassically
function calls and
determine
f (0) calls
and f (1)
to decide
since we
need= f(1)?
two
function
are
needed:
f(0)
whether f (0) = f (1). Equivalently we can calculate f (0) ⊕ f (1), where ⊕
nary addition modulo 2. If this value is zero, then f is constant.
unction f is given as a quantum algorithm Uf that takes an input state |x⟩|y⟩
▪ Quantum mechanically only one function call by applying
|x⟩|f (x) ⊕ y⟩ then we can determine whether the function is constant in a
to both
arguments at once
ction callfunction
with the following
algorithm:
|0⟩
H
|1⟩
H
U f x y → x f (x) ⊕ y
H
Uf
the
U f (α 0 + β 1 ) 0 → α 0 f (0) + β 1 f (1)
tart in a state |0⟩|1⟩ and apply a Hadamard gate to each qubit, giving the
+ |1⟩)(|0⟩ − |1⟩). Applying the function f we obtain
needed
to read out the answer as one bit
▪ Smart manipulation
"
1!
(−1)f (0)
|0⟩ + (−1)1f (0)⊕f (1) |1⟩ (|0⟩ − |1⟩).
(8.7)
2
⎡ 1+ (−1) f (0)⊕ f (1) 0 + 1− (−1) f (0)⊕ f (1) 1 ⎤
⎦
2⎣
state of the second qubit is constant and the global phase irrelevant we drop
and focus on just the first qubit’s state √1 (|0⟩ + (−1)f (0)⊕f (1) |1⟩). Applying
(
)
(
)
Matthias Troyer
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What would we use it for?
!
!
Surprisingly, this question has not been seriously asked until 2012
!
It needs to be an important problem (killer-app) that we cannot solve on an exascale machine
!
Compare to state-of-the-art on beyond-exascale hardware and not an unoptimized code on a single CPU core
Matthias Troyer
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Grover search
▪ Search an unsorted database of N entries in √N time
▪ Rare case of provable quantum speedup given an oracle
▪ However, the oracle needs to be implemented!
▪ N-entry database or arbitrary stored data needs at least O(N) hardware
▪ Can perform the same search classically in log(N) time with special purpose hardware
▪ Grover search is only useful if the database need not be stored but can be calculated on the fly!
Matthias Troyer
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Shor’s algorithm for factoring
▪ Factoring is a hard problem classically: O(exp(N1/3)) for N bits
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63096709107717143027979502211512024167962284944780565098736835024782968305430921627667450973510563924
02989775917832050621619158848593319454766098482875128834780988979751083723214381986678381350567167 !
=
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9135372537443549282380180405548453067960658656053548608342707327969894210413710440109013191728001673!
* !
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6058795416957989730803763009700969113102979143329462235916722607486848670728527914505738619291595079
▪ Shor’s algorithm is polynomial time on a quantum computer
▪ O(N3) using minimal number of 2N+3 qubits
▪ O(N2) using O(N) qubits
▪ O(N) using O(N2) qubits
▪ Shor’s algorithm suddenly made quantum computing interesting
Matthias Troyer
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Shor’s algorithm and encryption
▪ Shor’s algorithm can be used to crack RSA encryption
▪
▪
assuming 10 ns gate time and minimal number of 2N+3 qubits
much faster (seconds) when using more qubits
RSA
cracked in
CPU years
Shor
453 bits
1999
10
1 hour
768 bits
2009
2000
5 hours
1000000
10 hours
1024 bits
▪ But use of quantum computers to crack RSA is limited since we can
anytime switch to post-quantum encryption
▪ quantum cryptography
▪ lattice based cryptography
Matthias Troyer
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Solving linear systems of equations (Harrow et al)
▪ Solve linear systems in log(N) time if
▪ matrix can be computed efficiently and need not be stored
▪ only log (N) bits of the answer are needed
▪ problem is well conditioned
▪ First application: electromagnetic wave scattering
▪ Clader, Jacobs, Sprouse (2013)
▪ crossover compared to classical supercomputers is beyond 1000 years wallclock time
lx
ly
Scattered Field
Incident Field
Matthias Troyer
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Applications running at scale on Jaguar @ ORNL
Applications running at scale on Jaguar @ ORNL (Spring 2011)
Domain area
Code name
Institution
# of cores
Performance
Notes
Materials
DCA++
ORNL
213,120
1.9 PF
PF
1.9
2008 Gordon Bell
Prize Winner
Materials
WL-LSMS
ORNL/ETH
223,232
1.8 PF
2009 Gordon Bell
Prize Winner
Chemistry
NWChem
PNNL/ORNL
224,196
1.4 PF
2008 Gordon Bell
Prize Finalist
Materials
DRC
ETH/UTK
186,624
1.3 PF
2010 Gordon Bell
Prize Hon. Mention
Nanoscience
OMEN
Duke
222,720
> 1 PF
2010 Gordon Bell
Prize Finalist
Biomedical
MoBo
GaTech
196,608
780 TF
2010 Gordon Bell
Prize Winner
Chemistry
MADNESS
UT/ORNL
140,000
550 TF
Materials
LS3DF
LBL
147,456
442 TF
2008 Gordon Bell
Prize Winner
149,784
165 TF
2008 Gordon Bell
Prize Finalist
147,456
83 TF
Seismology
SPECFEM3D USA (multiple)
Combustion
SNL
Source: S3D
T. Schulthess
Matthias Troyer
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Solving quantum chemistry on a quantum computer
▪ A killer-app for quantum computing is solving quantum problems
▪ Design a room-temperature superconductor
▪ Develop a catalyst for carbon sequestration
▪
Develop better catalysts for nitrogen fixation (fertilizer)
▪ These problem need better accuracy than we get by using
approximate classical algorithms
▪ exponentially hard classically
▪ polynomial complexity on quantum hardware
Matthias Troyer
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Quantum algorithms for quantum chemistry
▪ Assume the fastest imaginable quantum computer and consider smallest
problem we cannot solve classically: N=50 orbitals, O(100) qubits
Scaling
Run time for N=50
State of 12/2013
N10
100 000 years
1/2014 pipelining and
optimisations
6/2014 faster convergence
parallel with
with
N9
3 years
N5.5
1 hour
N1.6
seconds?
▪ Quantum information theorists declare victory proving the existence of
polynomial time algorithms
▪ We need HPQC specialists to develop better algorithms!
Matthias Troyer
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State of the art of hardware
!
!
!
many proposals but few are scalable …
Matthias Troyer
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Physical realizations and decoherence
▪ Coupling to the environment easily destroys the quantum superposition
▪ Qubits need to be designed to be well isolated from the environment
f
ψ =α 0 +β 1
i
Parallel evolution
providing the
quantum speedup.
Perturbations
from the environment
destroy the parallel evolution of the computation
21
Matthias Troyer
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The best qubits: ion traps (universities)
▪ Use the motional states of well isolated ions to encode a qubit
▪
▪
▪
▪
up to 14 qubits
coherent for several seconds
about 100 gate operations
10 µs gate times
Image: group of R. Blatt, Innsbruck
!
▪ Advantages and disadvantages
▪ Well isolated from environment
▪ Relatively slow
▪ Hard to scale beyond O(20) qubits
Matthias Troyer
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Superconducting qubits (IBM and universities)
▪ Use superconducting current loops to encode a qubit
▪ State of the art
▪
▪
▪
▪
5 qubits
100 µs coherence times
10 ns gate times
about 100 gate operations
▪ Advantages and disadvantages
▪
▪
▪
▪
scalable
fast
can be built in semiconductor foundries
lots of decoherence in a solid state device
Matthias Troyer
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Quantum error correction
▪ Example: Shor’s code protects a against sign or bit flip errors
▪ Needs nine qubits and substantially increases number of gates
▪ Codes need to be recursively iterated to protect against multiple errors
▪ Quantum error correction overhead is at least 1000x more qubits and
gates for any reasonable computation
Matthias Troyer
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Topological qubits (Microsoft and universities)
▪ Encode the qubit in a topological property of a quantum state
▪
▪
▪
▪
Intrinsic protection due to topology
No local noise can decohere the qubit
No expensive error correction needed
Operations done by “braiding” isolated particles
▪ Most promising: Majorana particles
▪ may exist at ends of superconducting nanowires
▪ hope to confirm their existence within 5 years
Ettore
Majorana
Matthias Troyer
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(Optimistic) road map for quantum computers
§ First quantum devices exist, but computational power is limited
§ quantum random numbers
§ quantum encryption
§ quantum simulators
§ quantum optimizers
§ General-purpose quantum computers will always remain
special purpose accelerators for selected tasks
§ My optimistic road map
§ Today: 14 qubits allowing up to 100 operations
§ 2020: a long-time stable quantum bit with quantum error correction
§ 2025: matching classical CPUs on selected applications
§ 2030: outperforming any hypothetical classical computer on selected applications
§ Huge potential for classical spin-off technology (e.g. superconducting)
Matthias Troyer
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26
Quantum computing applications
▪ Identifying killer-apps for quantum computing is challenging
▪
▪
▪
▪
▪
the problem has to be hard enough that it cannot be solved on a PC
the problem has to be amenable to quantum acceleration
the crossover scale has to be short enough to make it useful
CMOS technology is a tough competitor
we need to consider special purpose classical devices as competitors
▪ Potential applications
▪
▪
▪
▪
factoring and code breaking (limited use)
quantum chemistry and material simulations (challenging but enormous potential)
solving linear systems (can’t we solve them well enough already?)
machine learning???
▪ It is time to move quantum computing research
from theoretical computer science to high performance computing
Matthias Troyer
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