Improving Gate-Level Simulation of Quantum Circuits

George F. Viamontes,

Igor L. Markov, and John P. Hayes{gviamont,imarkov,jhayes}@umich.edu

Advanced Computer Architecture Laboratory

University of Michigan, EECS

DARPADARPA

Problem

• Simulation of quantum computing on a classical computer– Requires exponentially growing time and

memory resources

• Goal: Improve classical simulation

• Our Solution: Quantum Information Decision Diagrams (QuIDDs)

Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Ongoing Work

Quantum Data

• Classical bit– Two possible states: 0 or 1– Measurement is straightforward

• Qubit (properties follow from Q. M.)– Quantum state– Can be in states 0 or 1, but also

in a superposition of 0 and 1– n qubits represents

different values simultaneously– Measurement is probabilistic and destructive

n2

Implementations

• Liquid and solid state nuclear magnetic resonance (NMR) – nuclear spins

• Ion traps – electron energy levels• Electrons floating on liquid helium –

electron spins• Optical technologies – photon polarizations• Focus of this work:

common mathematical description

Qubit Notation

• Qubits expressed in Dirac notation

• Vector representation:

• and are complex numbers called probability amplitudes s.t.

10

1|||| 22

Data Manipulation

• Qubits are manipulated by operators– Analogous to logic gates

• Operators are unitary matrices

• Matrix-vector multiplication describes operator functionality

U ' U '

Operations on Multiple Qubits

• Tensor product of operators/qubits

2/12/12/12/1

2/12/12/12/1

2/12/12/12/1

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2/12/1

2/12/1

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2/12/1

H '

H ' '' HH

Previous Work• Traditional array-based representations

are insensitive to the values stored• Qubit-wise multiplication

– 1-qubit operator and n-qubit state vector– State vector requires exponential memory

• BDD techniques– Multi-valued logic for q. circuit synthesis [1]– Shor’s algorithm simulator (SHORNUF) [8]

Redundancy in Quantum Computing

• Matrix/vector representation of quantum gates/state vectors contains block patterns

• The tensor product propagates block patterns in vectors and matrices

Example of Propagated Block Patterns

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Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Ongoing Work

Data Structure that Exploits Redundancy

• Binary Decision Diagrams (BDDs) exploit repeated sub-structure

• BDDs have been used to simulate classical logic circuits efficiently [6,2]

• Example: f = a AND ba

f

b

1 0

Assign value of 1 to variable x

Assign value of 0 to variable x

BDDs in Linear Algebra

• Algebraic Decision Diagrams (ADDs) treat variable nodes as matrix indices [2], also MTBDDs

• ADDs encode all matrix elements aij

– Input variables capture bits of i and j– Terminals represent the value of aij

• CUDD implements linear algebra for ADDs (without decompression)

Quantum Information Decision Diagrams (QuIDDs)

• QuIDDs: an application of ADDsto quantum computing

• QuIDD matrices : row (i), column (j) vars• QuIDD vectors: column vars only• Matrix-vector multiplication

performed in terms of QuIDDs

1 1i j

jijij bac

QuIDD Vectorsf

0C

1C

1 0

1

0

0

0

00

01

10

11

Terminal value array

0 + 0i

1 + 0i

0

1

11

QuIDD Matrices

i02/1

i02/1

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01

10

11

1001 1100

1

f

0R

0C

1R

1C

1R

1C

01

0

QuIDDs and ADDs

• All dimensions are 2n

• Row and column variables are interleaved

• Terminals are integers which map into an array of complex numbers

TCRCRCR nn 1100

Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Conclusions

QuIDD Operations

• Based on the Apply algorithm [4,5]

– Construct new QuIDDs by traversing two QuIDD operands based on variable ordering

– Perform “op” when terminals reached (op is *, +, etc.)

– General Form: f op g where f and g are QuIDDs, and x and y are variables in f and g, respectively:

ix

ii yx gopfii yx

gopf

iy

iyi gopxiyi gopx

ix

ix yopfi ix

yopfi

ii yx ii yx ii yx

Tensor Product

• Given A B– Every element of a matrix A

is multiplied by the entire matrix B

• QuIDD Implementation: Use Apply– Operands are A and B – Variables of operand B are shifted– “op” is defined to be multiplication

Other Operations

• Matrix multiplication– Modified ADD matrix multiply algorithm [2]– Support for terminal array– Support for row/column variable ordering

• Matrix addition– Call to Apply with “op” set to addition

• Qubit measurement– DFS traversal or measurement operators

Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Ongoing Work

H

H

H

Oracle

H

H

Conditional Phase Shift

H

H

H

|0>

|0>

|1>

.

.

.

.

.

.

.

.

.

Grover’s Algorithm

- Search for items in an unstructured database of N items - Contains n = log N qubits and has runtime NO

Number of Iterations

• Use formulation from Boyer et al. [3]

• Exponential runtime(even on an actual quantum computer)

• Actual Quantum Computer Performance: ~ O(1.41n) time and O(n) memory

Simulation Results forGrover’s Algorithm

• Linear memory growth(numbers of nodes shown)

Results: Oracle 1

Linear Growth using QuIDDPro

Oracle 1: Runtime (s)

No. Qubits (n) Octave MATLAB Blitz++ QuIDDPro

10 80.6 6.64 0.15 0.33

11 2.65e2 22.5 0.48 0.54

12 8.36e2 74.2 1.49 0.83

13 2.75e3 2.55e2 4.70 1.30

14 1.03e4 1.06e3 14.6 2.01

15 4.82e4 6.76e3 44.7 3.09

16 > 24 hrs > 24 hrs 1.35e2 4.79

17 > 24 hrs > 24 hrs 4.09e2 7.36

18 > 24 hrs > 24 hrs 1.23e3 11.3

19 > 24 hrs > 24 hrs 3.67e3 17.1

20 > 24 hrs > 24 hrs 1.09e4 26.2

Oracle 1: Peak Memory Usage (MB)

No. Qubits (n) Octave MATLAB Blitz++ QuIDDPro

10 2.64e-2 1.05e-2 3.52e-2 9.38e-2

11 5.47e-2 2.07e-2 8.20e-2 0.121

12 0.105 4.12e-2 0.176 0.137

13 0.213 8.22e-2 0.309 0.137

14 0.426 0.164 0.559 0.137

15 0.837 0.328 1.06 0.137

16 1.74 0.656 2.06 0.145

17 3.34 1.31 4.06 0.172

18 4.59 2.62 8.06 0.172

19 13.4 5.24 16.1 0.172

20 27.8 10.5 32.1 0.172

Linear Growth using QuIDDPro

Validation of Results

• SANITY CHECK: Make sure that QuIDDPro achieves highest probability of measuring the item(s) to be searched using the number of iterations predicted by Boyer et al. [3]

Consistency with Theory

Grover Results Summary

• Asymptotic performance– QuIDDPro: ~ O(1.44n) time and O(n) memory– Actual Quantum Computer

• ~ O(1.41n) time and O(n) memory

• Outperforms other simulation techniques– MATLAB: (2n) time and (2n) memory– Blitz++: (4n) time and (2n) memory

What about errors?

• Do the errors and mixed states that are encountered in practical quantum circuits cause QuIDDs to explode and lose significant performance?

NIST Benchmarks

• NIST offers a multitude of quantum circuit descriptions containing errors/decoherence and mixed states

• NIST also offers a density matrix C++ simulator called QCSim

• How does QuIDDPro compare to QCSim on these circuits?

QCSim vs. QuIDDPro

• dsteaneZ: 13-qubit circuit with initial mixed state that implements the Steane code to correct phase flip errors– QCSim: 287.1 seconds, 512.1MB– QuIDDPro: 0.639 seconds, 0.516 MB

QCSim vs. QuIDDPro (2)

• dsteaneX: 12-qubit circuit with initial mixed state that implements the Steane code to correct bit flip errors– QCSim: 53.2 seconds, 128.1MB– QuIDDPro: 0.33 seconds, 0.539 MB

Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Ongoing Work

Recall the Tensor Product

Key Formula

• Given QuIDDs , the tensor product QuIDD contains

nodes

niiQ 1}{

ini Q1

n

i

iji TermQInQIn

2

111 |)(||)(||)(|

|)(| 1 ini QTerm

Persistent Sets

• A set is persistent if and only if the set of n pair-wise products of its elements is constant (i.e. the pair-wise product n times)

• Consider the tensor product of two matrices whose elements form a persistent set– The number of unique elements in the resulting

matrix will be a constant with respect to the number of unique elements in the operands

Relevance to QuIDDs

• Tensor products with n QuIDDs whose terminals form a persistent set produce QuIDDs whose sets of terminals do not increase with n

Main Results

• Given a persistent set and a constant C, consider n QuIDDs with at most C nodes each and terminal values from . The tensor product of those QuIDDs has O(n) nodes and can be computed in O(n) time.

• Matrix multiplication with QuIDDs A and B as operands requires time and produces a result with nodes [2]

))(( 2ABO))(( 2ABO

Applied to Grover’s Algorithm

• Since O(1.41n) Grover iterations are required, and thus O(1.41n) matrix multiplications, does Grover’s algorithm induce exponential memory complexity when using QuIDDs?

• Answer: NO! – The internal nodes of the state vector/density matrix

QuIDD is the same at the end of each Grover iteration

– Runtime and memory requirements are therefore polynomial in the size of the oracle QuIDD

Outline

• Background

• QuIDD Structure

• QuIDD Operations

• Simulation Results

• Complexity Analysis Results

• Ongoing Work

Ongoing Work

• Explore error/decoherence models• Simulate Shor’s algorithm

– QFT and its inverse are exponential in size as QuIDDs

– Other operators are linear in size as QuIDDs– QFT and its inverse are an asymptotic

bottleneck

• Limitations of quantum computing

Relevant Work

G. Viamontes, I. Markov, J. Hayes, “Improving Gate-Level Simulation of Quantum circuits,” Los Alamos Quantum Physics Archive, Sept. 2003 (quant-ph/0309060)

G. Viamontes, M. Rajagopalan, I. Markov, J. Hayes, “Gate-Level Simulation of Quantum Circuits,” Asia South Pacific Design Automation Conference, pp. 295-301, January 2003

G. Viamontes, M. Rajagopalan, I. Markov, J. Hayes, ‘Gate-Level Simulation of Quantum Circuits,” 6th Intl. Conf. on Quantum Communication, Measurement, and Computing, pp. 311-314, July 2002

References

[1] A. N. Al-Rabadi et al., “Multiple-Valued Quantum Logic,” 11th Intl. Workshop on Post Binary ULSI, Boston, MA, May 2002.

[2] R. I. Bahar et al., “Algebraic Decision Diagrams and their Applications”, In Proc. IEEE/ACM ICCAD, pp. 188-191, 1993.

[3] M. Boyer et al., “Tight Bounds on Quantum Searching”, Fourth Workshop on Physics and Computation, Boston, Nov 1996.

[4] R. Bryant, “Graph-Based Algorithms for Boolean Function Manipulation”, IEEE Trans. On Computers, vol. C-35, pp. 677-691, Aug 1986.

[5] E. Clarke et al., “Multi-Terminal Binary Decision Diagrams and Hybrid Decision Diagrams”, In T. Sasao and M. Fujita, eds, Representations of Discrete Functions, pp. 93-108, Kluwer, 1996.

References

[6] C.Y. Lee, “Representation of Switching Circuits by Binary Decision Diagrams,” Bell System Technical Jour., 38:985-999, 1959.

[7] D. Gottesman, “The Heisenberg Representation of Quantum Computers,” Plenary Speech at the 1998 Intl. Conf. on Group Theoretic Methods in Physics, http://xxx.lanl.gov/abs/quant-ph/9807006

[8] D. Greve, “QDD: A Quantum Computer Emulation Library,” http://home.plutonium.net/~dagreve/qdd.html