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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 DARPA 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 2 n different values simultaneously – Measurement is probabilistic and destructive 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 0 1 • Vector representation: • and are complex numbers called probability amplitudes s.t. | |2 | |2 1 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 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 H ' H ' 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 H H 1/ 2 1 / 2 1 / 2 1/ 2 ' ' 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 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1/ 2 1/ 2 1/ 2 1/ 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 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] f • Example: f = a AND b a Assign value of 1 to variable x b Assign value of 0 to variable x 1 0 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 ADDs to quantum computing • QuIDD matrices : row (i), column (j) vars • QuIDD vectors: column vars only • Matrix-vector multiplication cij aijb j i 1 j 1 performed in terms of QuIDDs QuIDD Vectors f C0 C1 1 0 0 0 + 0i 1 1 + 0i 0 00 0 01 0 10 11 1 11 Terminal value array QuIDD Matrices f 00 R0 C0 R1 R1 00 01 10 11 C1 C1 1 10 11 1/ 2 1/ 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 0 0 01 1 / 2 0i 1 1 / 2 0i QuIDDs and ADDs • All dimensions are 2n • Row and column variables are interleaved R0 C0 R1 C1 Rn Cn T • Terminals are integers which map into an array of complex numbers 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: xi yi xi yi xi yi xi yi xi f xi op g yi f xi op g yi xi op g yi xi op g yi f xi op yi f xi op yi 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 Grover’s Algorithm |0> H H H |0> H H H . . . |1> Oracle . . . Conditional Phase Shift . . . H H - Search for items in an unstructured database of N items - Contains n = log N qubits and has runtime O N 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 for Grover’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 n i i 1 • Given QuIDDs {Q } , the tensor product n QuIDD i 1 Qi contains | In(Q1 ) | i 2 | In(Qi ) || Term( ) | n | Term(in1 Qi ) | nodes i 1 j 1 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 O(( AB) 2time ) and produces a result with O(( AB) 2 ) nodes [2] 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, “GateLevel Simulation of Quantum Circuits,” Asia South Pacific Design Automation Conference, pp. 295-301, January 2003 G. Viamontes, M. Rajagopalan, I. Markov, J. Hayes, ‘GateLevel 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. 677691, 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/quantph/9807006 [8] D. Greve, “QDD: A Quantum Computer Emulation Library,” http://home.plutonium.net/~dagreve/qdd.html