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A Compiler Writer Looks at Quantum Computation
1. Why is there so much excitement about quantum computation?
2. Computational models for quantum programming
3. Candidate target-machine technologies
4. Quantum programming languages
5. Unsolved issues in quantum computation
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What the Physicists are Saying
“Quantum information is aradical departure in informationtechnology, more fundamentallydifferent from current technologythan the digital computer is fromthe abacus.”
William D. Phillips,
1997 Nobel Prize Winner in Physics
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Shor’s Integer Factorization Algorithm
Problem: Given a composite n-bit integer, find a nontrivial factor.
Best-known deterministic algorithm on a classical computer has time complexity exp(O( n1/3 log2/3 n )).
A quantum computer can solve thisproblem in O( n3 ) operations.
Peter ShorAlgorithms for Quantum Computation: Discrete Logarithms and Factoring
Proc. 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124-134
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Integer Factorization: Estimated Times
Classical: number field sieve–Time complexity: exp(O(n1/3 log2/3 n))–Time for 512-bit number: 8400 MIPS years–Time for 1024-bit number: 1.6 billion times longer
Quantum: Shor’s algorithm–Time complexity: O(n3)–Time for 512-bit number: 3.5 hours–Time for 1024-bit number: 31 hours
(assuming a 1 GHz quantum device)
M. Oskin, F. Chong, I. ChuangA Practical Architecture for Reliable Quantum Computers
IEEE Computer, 2002, pp. 79-87
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Computational Thinking
Computational thinking is a fundamental skill for everyone, not just for computer scientists. To reading, writing, and arithmetic, we should add computational thinking to every child’s analytical ability. Just as the printing press facilitated the spread of the three Rs, what is appropriately incestuous about this vision is that computing and computers facilitate the spread of computational thinking.
Jeannette M. WingComputational Thinking
CACM, vol. 49, no. 3, pp. 33-35, 2006
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What is Computational Thinking?
The thought processes involved in formulating problems so their solutions can be represented as computation steps and algorithms.
Alfred V. AhoComputation and Computational Thinking
The Computer Journal, vol. 55, no. 7, pp. 832- 835, 2012
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Models of Computation in Languages
Underlying most programming languages is a model of computation:
Procedural: Fortran, C
Functional: Lisp, ML, OCaml, Haskell
Object oriented: C++, Java
Logic: Prolog
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Three Models of Computation forQuantum Computing
1. Quantum circuits
2. Topological quantum computing
3. Adiabatic quantum computing
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Towards a Model of Computation forQuantum Computing
TargetSystem
MathematicalFormulation
Algorithms
Model ofComputation
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The Mathematical Underpinnings ofQuantum Computing
The Four Postulates of Quantum Mechanics
M. A. Nielsen and I. L. ChuangQuantum Computation and Quantum Information
Cambridge University Press, 2000
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State Space Postulate
The state of an isolated quantum system can be describedby a unit vector in a complex Hilbert space.
Postulate 1
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Qubit: Quantum Bit
• The state of a quantum bit in a 2-dimensional complex Hilbert space can be described by a unit vector (in Dirac notation)
where α and β are complex coefficients called the amplitudes of the basis states |0i and |1i and
• In conventional linear algebra
10
a b
a = ba = ba1 = b1
122
1
0
0
1
1
0
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Time-Evolution Postulate
Postulate 2
The evolution of a closed quantum systemcan be described by a unitary operator U.
(An operator U is unitary if U y = U −1.)
U U
state ofthe systemat time t1
state ofthe systemat time t2
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Useful Quantum Operators: Pauli Operators
Pauli operators
10
01
0
0
01
10
10
01Z
i
iYXI
X0 1
In conventional linear algebrais equivalent to
1
0
0
1
01
10
10 X
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Useful Quantum Operators: Hadamard Operator
The Hadamard operator has the matrix representation
H maps the computational basis states as follows
Note that HH = I.
11
11
2
1H
)10(2
11
)10(2
10
H
H
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Composition-of-Systems Postulate
1
The state space of a combined physical system is
the tensor product space of the state spaces of the
component subsystems.
If one system is in the state and another is in
the state , then the combined system is in the
state .
is often written as or as .
Postulate 3
221
21 21 21
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Useful Quantum Operators: the CNOT Operator
The two-qubit CNOT (controlled-NOT) operator:
CNOT flips the target bit t iff the control bit c has the value 1:
0100
1000
0010
0001
.c
t
c
tc
The CNOT gate maps
1011,1110,0101,0000
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Measurement Postulate
Quantum measurements can be described by a
collection of operators acting on the state space
of the system being measured. If the state of the
system is before the measurement, then the
probability that the result m occurs is
and the state of the system after measurement is
Postulate 4
}{ mM
mm MMmp †)(
|| †mm
m
MM
M
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Measurement
The measurement operators satisfy the completeness equation:
The completeness equation says the probabilities sum to one:
IMM mm
m †
1)( † MMmp mm m
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Quantum Circuit Model for Quantum Computation
Quantum circuit to create Bell (Einstein-Podulsky-Rosen) states:
Circuit maps
Each output is an entangled state, one that cannot be written in a product form. (Einstein: “Spooky action ata distance.”)
x
y
2
)1001(11,
2
)1100(10,
2
)1001(01,
2
)1100(00
Hxy
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Alice and Bob’s Qubit-State Delivery Problem
• Alice knows that she will need to send to Bob the state of an important secret qubit sometime in the future.
• Her friend Bob is moving far away and will have a very low bandwidth Internet connection.
• Therefore Alice will need to send her qubit state to Bob cheaply.
• How can Alice and Bob solve their problem?
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Alice and Bob’s Solution: Quantum Teleportation!
• Alice and Bob generate an EPR pair.
• Alice takes one half of the pair; Bob the other half. Bob moves far away.
• Alice interacts her secret qubit with her EPR-half and measures the two qubits.
• Alice sends the two resulting classical measurement bits to Bob.
• Bob decodes his half of the EPR pair using the two bits to discover .
00
H
X Z
M1
M2
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Quantum Computer Architecture
Knill [1996]: Quantum RAM, a classical computer combined with a quantum device with operations for initializing registers of qubits and applying quantum operations and measurements
QuantumMemory
QuantumLogic Unit
Classical Computer
E. KnillConventions for Quantum Pseudocode
Los Alamos National Laboratory, LAUR-96-2724, 1996
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Cross’s Fault-TolerantQuantum Computer Architecture
QuantumMemory
QuantumLogic Unit
Classical Computer
AncillaFactory
QuantumSoftwareFactory
0
Andrew W. CrossFault-Tolerant Quantum Computer Architectures
Using Hierarchies of Quantum Error-Correcting CodesPhD Thesis, MIT, June 2008
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DiVincenzo Criteria for a Quantum Computer
1. Be a scalable system with well-defined qubits
2. Be initializable to a simple fiducial state
3. Have long decoherence times
4. Have a universal set of quantum gates
5. Permit efficient, qubit-specific measurements
David DiVincenzoThe Physical Implementation of Quantum Computation
arXiv:quant-ph/0002077v3
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Candidate Target-Machine Technologies
• Ion traps
• Josephson junctions
• Nuclear magnetic resonance
• Optical photons
• Optical cavity quantum electrodynamics
• Quantum dots
• Nonabelian fractional quantum Hall effect anyons
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MIT Ion Trap Simulator
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Ion Trap Quantum Computer: The Reality
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Universal Sets of Quantum Gates
A set of gates is universal for quantum computation if any unitary operation can be approximated to arbitrary accuracy by a quantum circuit using gates from that set.
The phase-gate S = ; the π/8 gate T =
Common universal sets of quantum gates:• { H, S, CNOT, T }• { H, I, X, Y, Z, S, T, CNOT }
CNOT and the single qubit gates are exactly universal for quantum computation.
i0
01
4/0
01ie
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Shor’s Quantum Factoring Algorithm
Input: A composite number NOutput: A nontrivial factor of N
if N is even then return 2;if N = ab for integers a >= 1, b >= 2 then return a;
x := rand(1,N-1);if gcd(x,N) > 1 then return gcd(x,N);r := order(x mod N); // only quantum stepif r is even and xr/2 != (-1) mod N then {f1 := gcd(xr/2-1,N); f2 := gcd(xr/2+1,N)};
if f1 is a nontrivial factor then return f1;else if f2 is a nontrivial factor then return f2;else return fail;
Nielsen and Chuang, 2000
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The Order-Finding Problem
Given positive integers x and N, x < N, such thatgcd(x, N) = 1, the order of x (mod N) is the smallest positive integer r such that xr ≡ 1 (mod N).
E.g., the order of 5 (mod 21) is 6.
The order-finding problem is, given two relatively prime integers x and N, to find the order of x (mod N).
All known classical algorithms for order finding aresuperpolynomial in the number of bits in N.
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Quantum Order Finding
Order finding can be done with a quantum circuit containing
O((log N)2 log log (N) log log log (N))
elementary quantum gates.
Best known classical algorithm requires
exp(O((log N)1/2 (log log N)1/2 )
time.
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Some Proposed Quantum Programming Languages
•Quantum pseudocode [Knill, 1996]
• Imperative: e.g., QCL [Ömer, 1998-2003]–syntax derived from C
–classical flow control
–classical and quantum data
–interleaved measurements and quantum operators
•Functional: e.g., Quipper [Green et al., 2013]–strongly typed functional programming language
–uses Haskell as the host language
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Language Abstractions and Constraints
• States are superpositions
• Operators are unitary transforms
• States of qubits can become entangled
• Measurements are destructive
• No-cloning theorem: you cannot copy an unknown quantum state!
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Quantum Algorithm Design Techniques
• Phase estimation
• Quantum Fourier transform
• Period finding
• Eigenvalue estimation
• Grover search
• Amplitude amplification
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Quantum Computer Design Tools: Desiderata• A design flow that will map high-level quantum programs into
efficient fault-tolerant technology-specific implementations on different quantum computing devices
• Languages, compilers, simulators, and design tools to support the design flow
• Well-defined interfaces between components
• Efficient methods for incorporating fault tolerance and quantum error correction
• Efficient algorithms for optimizing and verifying quantum programs
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Quantum Design Tools Hierarchy
• Vision: Layered hierarchy with well-defined interfaces
Programming Languages
Compilers
Optimizers Layout Tools Simulators
K. Svore, A. Aho, A. Cross, I. Chuang, I. MarkovA Layered Software Architecture for Quantum Computing Design Tools
IEEE Computer, 2006, vol. 39, no. 1, pp. 74-83
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Languages and Abstractions in the Design Flow
FrontEnd
TechnologyIndependent
CG+Optimizer
TechnologyDependent
CG+Optimizer
TechnologySimulator
quantumsource
program
QIR QASM QPOL
QIR: quantum intermediate representationQASM: quantum assembly languageQPOL: quantum physical operations language
quantumcircuit
quantumcircuit
quantumdevice
quantummechanics
ABSTRACTIONS
Quantum Computer Compiler
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Design Flow for Ion Trap
Mathematical Model:Quantum mechanics,
unitary operators,tensor products
Physical Device
Computational Formulation:
Quantum bits, gates, and circuits
TargetQPOL
Physical System:Laser pulses
applied to ions in traps
Quantum Circuit ModelEPR Pair Creation QIR QPOLQASM
QCC:QIR,
QASM
Machine Instructions
A 21 3
A 21 3
B
B
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Fault Tolerance
• In a fault-tolerant quantum computer, more than 99% of the resources spent will probably go to quantum error correction [Chuang, 2006].
• A circuit containing N (error-free) gates can be simulated with probability of error at most ε, using N log(N/ε) faulty gates, which fail with probability p, so long as p < pth [von Neumann, 1956].
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Fault Tolerance
• Obstacles to applying classical error correction to quantum circuits:–no cloning–errors are continuous–measurement destroys information
• Shor [1995] and Steane [1996] showed that these obstacles can be overcome with concatenated quantum error-correcting codes.
P. W. ShorScheme for Reducing Decoherence in Quantum Computer Memory
Phys. Rev. B 61, 1995
A. SteaneError Correcting Codes in Quantum Theory
Phys. Rev. Lett. 77, 1996
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Mathematical Model:Quantum mechanics,
unitary operators,tensor products
Computational Formulation:
Quantum bits, gates, and circuits
Software:QPOL
Physical System:Laser pulses
applied to ions in traps
Quantum Circuit ModelEPR Pair Creation QIR QPOLQASM
QCC:QIR,
QASM
Machine Instructions Physical Device
A 21 3
A 21 3
B
B
Design Flow with Fault Tolerance andError Correction
Fault Tolerance and Error Correction (QEC)
QEC
QEC
Moves Moves
K. SvorePhD Thesis
Columbia, 2006
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Recent work:Synthesis and simulation of quantum circuits
Synthesis of efficient quantum circuits
• faster phase estimation [Svore, Hastings & Freedman, 2013]
• depth-optimal single-qubit circuits [Bocharov & Svore, 2012]
• fault-tolerant single-qubit rotations [Duclos-Cianci & Svore, 2012]
• fast synthesis of depth-optimal quantum circuits [Amy, Maslov, Mosca & Roetteler, 2012]
• exact synthesis of multi-qubit Clifford and T- circuits [Giles & Sellinger, 2012]
Efficient simulation of quantum circuits
• QuIDDPro quantum circuit simulator [Viamontes, Markov & Hayes,University of Michigan, 2009]
• LIQUi|> software architecture and toolsuite [Wecker, Microsoft Research, ongoing]
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S. Simon, N. Bonesteel, M. Freedman, N. Petrovic, and L. HormoziTopological Quantum Computing with Only One Mobile Quasiparticle
Phys. Rev. Lett, 2006
A Second Model for Quantum Computing:Topological Quantum Computing
In any topological quantum computer, all computations can be performed by moving only a single quasiparticle!
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Topological Robustness
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Topological Robustness
=
=tim
e
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Bonesteel, Hormozi, Simon, … ; PRL 2005, 2006; PRB 2007
U
U
Quantum Circuit
time
Braid
=
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C. Nayak, S. Simon, A. Stern, M. Freedman, S. DasSarmaNon-Abelian Anyons and Topological Quantum Computation
Rev. Mod. Phys., June 2008
1. Degenerate ground states (in punctured system) act as the qubits.
2. Unitary operations (gates) are performed on ground state by braiding punctures (quasiparticles) around each other.
Particular braids correspond to particular computations.
3. State can be initialized by “pulling” pairs from vacuum. State can be measured by trying to return pairs to vacuum.
4. Variants of schemes 2,3 are possible.
Advantages:
• Topological Quantum “memory” highly protected from noise• The operations (gates) are also topologically robust
Kitaev Freedman
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Universal Set of Topologically Robust Gates
U
USingle qubit rotations:
Controlled NOT:
Bonesteel, Hormozi, Simon, 2005, 2006
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Target Code Braid for CNOT Gatewith Solovay-Kitaev optimization
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A Third Model for Quantum Computing:Adiabatic Quantum Computing
A quantum system will stay near its
instantaneous ground state if the Hamiltonian
that governs its evolution varies slowly
enough.
E. Fahri, J. Goldstone, S. Gutmann, M. SipserQuantum Computation by Adiabatic Evolution
arXiv:quant-ph/0001106
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D-Wave Systems Quantum Computer
• D-Wave Systems has announced a 128-qubit quantum computer called Rainier
• It uses superconducting quantum interference devices (SQUIDS) to store the qubits
• Computation is controlled by a framework of switches formed from Josephson junctions
• Processor is housed in a 10’x10’x10’ refrigerator at 20mK
http://www.dwavesys.com/en/dev-tutorial-hardware.html
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Programming the D-Wave System
• The D-Wave System is designed to solve discrete optimization problems by finding the ground state of an instance of the Ising spin glass model
• “Programming at the machine language level is extremely difficult, even for our own internal developers.”
http://www.dwavesys.com/en/dev-tutorial-software.html
• A Python-based interface is provided. A programmer can specify an object function to be minimized. Machine returns a real number.
• Boixo et al. suggest the D-Wave system is a quantum annealer.
S. Boixo, T. Rønnow, S. Isakov, Z. Wang, D. Wecker, D. Lidar, J. Martinis, M. TroyerQuantum Annealing With More Than One Hundred Qubits
arXiv:1304.4595v2 [quant-ph]
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Example
• Consider the subset-sum problem: Does the set
{-7, -3, -2, 5, 8}
have a subset that sums to zero?
• Problem is formulated to minimize the generating function
where the variables are 0 or 1.
)1()85237(5
1
254321
k
kxxxxxx
http://www.dwavesys.com/en/dev-tutorial-software.html
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The D-Wave Systems Heuristic1. A series of random solutions are generated by the
conventional computing system.
2. The quality of these guesses is evaluated by passing them to the generating function.
3. The real numbers characterizing the goodness of the solutions are sent to the D-Wave system.
4. D-Wave automatically adjusts itself based on this feedback, and then generates a new series of guesses.
5. The new guesses are sent back to the conventional system where their goodness is evaluated.
6. Steps 3-5 are repeated until exit criteria are met.http://www.dwavesys.com/en/dev-tutorial-software.html
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Research Challenges
More qubits
Scalable, fault-tolerant architectures
Software tools
More algorithms!
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Collaborators
Andrew CrossSAIC
Igor MarkovU. Michigan
Krysta SvoreMicrosoft Research
Isaac ChuangMIT
TopologicalQuantum
Computing:Steve Simon
Oxford
