Projects on Quantum Circuits Simulation EECS Department University of Michigan, Ann Arbor, MI George F. Viamontes, Igor L. Markov, and John P. Hayes Advanced Computer Architecture Laboratory University of Michigan, EECS

Quantum Approaches to Logic Circuit Synthesis and Testing The University of Michigan: John Hayes and Igor Markov Quantum domain Classical domain Synthesis and test algorithms Automation Scalability Parallelism Microscale Practical logic circuits

New Ideas Integrated approach to both quantum and classical logic circuits Exploit quantum methods for classical circuit synthesis and testing problems Scalable representation of quantum circuits Multiobjective optimization techniques that are suited to automation (CAD)

Impact Quantum approaches promise more efficient computation in critical applications. Automation is key to large-scale quantum circuit development Development of practical algorithms for quantum circuit synthesis and test Use of quantum methods to improve testing and synthesis of classical circuits

Ongoing Projects at U.Michigan Simulation of quantum circuits –BDD-based QuIDDPro simulator –Simulating Grover’s algorithm Synthesis of two-qubit circuits –Bounds for gate counts in two-qubit circuits Quantum algorithms that improve memory usage –Quantum counters

Quantum Circuit Simulation Using QuIDDs Motivation –Need for a better way to simulate quantum circuits Quantum Information Decision Diagram (QuIDD) –Novel data representation that uses Binary Decision Diagrams (BDD) widely used in computer-aided circuit design –Captures some exponentially-sized matrices and vectors in a form that grows polynomially with the number of qubits –Multiplies matrices and vectors in compressed form QuIDDPro Simulator –Our QuIDD-based simulator implemented in C++ –Experiments with Grover’s algorithm demonstrate fast execution and low memory utilization

Problem Simulation of quantum computing on a classical computer –Requires exponentially growing time and memory resources Goal: Improve classical simulation Their Solution: Quantum Information Decision Diagrams (QuIDDs)

Operations on Multiple Qubits Tensor product of operators/qubits 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

QuIDD Structure

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 b a f b 10 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 a ij –Input variables capture bits of i and j –Terminals represent the value of a ij 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 performed in terms of QuIDDs

QuIDD Vectors f Terminal value array 0 + 0i 1 + 0i 0 1

QuIDD Data Representation f 1 0

QuIDDs and ADDs All dimensions are 2 n Row and column variables are interleaved Terminals are integers which map into an array of complex numbers

f 1 0 QuIDD Data Representation C0 = 1, R0=0, R1=1 R1,R0 \ C1, C0

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:

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

Simulation Results

QuIDDPro Simulation of Grover’s Algorithm H H H Oracle H H Conditional Phase Shift H H H |0> |1> Search for items in an unstructured database of N items - Contains n = log N qubits and has runtime

QuIDDPro Simulation Results (Grover’s search algorithm) for Oracle 1 Runtime Memory Usage Linear Growth using QuIDDPro 15 qubits Matlab Blitz++

QuIDDPro Simulation of Grover’s Algorithm Same results for any oracle that distinguishes a unique element Memory Usage Runtime

O(p(n)(√2) n ) O(n) QuIDDPro Simulation of Grover’s Algorithm Memory Usage Runtime

Number of Iterations Use formulation from Boyer et al. [3] Exponential runtime (even on an actual quantum computer) Actual Quantum Computer Performance: ~ O(1.41 n ) time and O(n) memory

Simulation Results for Grover’s Algorithm Linear memory growth (numbers of nodes shown)

Oracle 1: Runtime (s) No. Qubits (n)OctaveMATLABBlitz++QuIDDPro e e e32.55e e41.06e e46.76e > 24 hrs 1.35e > 24 hrs 4.09e > 24 hrs 1.23e > 24 hrs 3.67e > 24 hrs 1.09e426.2

Oracle 1: Peak Memory Usage (MB) No. Qubits (n)OctaveMATLABBlitz++QuIDDPro e-21.05e-23.52e-29.38e e-22.07e-28.20e e e 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(1.44 n ) time and O(n) memory –Actual Quantum Computer ~ O(1.41 n ) time and O(n) memory Outperforms other simulation techniques –MATLAB:  (2 n ) time and  (2 n ) memory –Blitz++:  (4 n ) time and  (2 n ) 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: seconds, 512.1MB –QuIDDPro: seconds, 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, MB

Complexity Analysis Results

Recall the Tensor Product

Key Formula Given QuIDDs, the tensor product QuIDD contains nodes

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]

Applied to Grover’s Algorithm Since O(1.41 n ) Grover iterations are required, and thus O(1.41 n ) 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 are the same at the end of each Grover iteration –Runtime and memory requirements are therefore polynomial in the size of the oracle QuIDD

Work in Progress: On The Power of Grover’s Algorithm Database search with a black-box predicate p(x)=1 –Classical evaluation of p(x) on one input (queries) –Quantum (parallel) evaluation of p(x) facilitates an implementation with fewer queries We also assume that p(x) is given as a BDD/QuIDD –BDDs are used to represent functions in practical CAD –However, a BDD is not really a black-box –BDD operations evaluate p(x) on multiple inputs at once (no quantum computation is involved) Grover on QuIDDs: same query complexity as in the quantum case –In practice this simulation is very fast and needs little memory Non-trivial assumption

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 (quant- ph/ ) G. Viamontes, M. Rajagopalan, I. Markov, J. Hayes, “Gate-Level Simulation of Quantum Circuits,” Asia South Pacific Design Automation Conference, pp , January 2003 G. Viamontes, M. Rajagopalan, I. Markov, J. Hayes, ‘Gate-Level Simulation of Quantum Circuits,” 6 th Intl. Conf. on Quantum Communication, Measurement, and Computing, pp , July 2002

References [1] A. N. Al-Rabadi et al., “Multiple-Valued Quantum Logic,” 11 th Intl. Workshop on Post Binary ULSI, Boston, MA, May [2] R. I. Bahar et al., “Algebraic Decision Diagrams and their Applications”, In Proc. IEEE/ACM ICCAD, pp , [3] M. Boyer et al., “Tight Bounds on Quantum Searching”, Fourth Workshop on Physics and Computation, Boston, Nov [4] R. Bryant, “Graph-Based Algorithms for Boolean Function Manipulation”, IEEE Trans. On Computers, vol. C-35, pp , Aug [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 , Kluwer, 1996.

References [6] C.Y. Lee, “Representation of Switching Circuits by Binary Decision Diagrams,” Bell System Technical Jour., 38: , [7] D. Gottesman, “The Heisenberg Representation of Quantum Computers,” Plenary Speech at the 1998 Intl. Conf. on Group Theoretic Methods in Physics, ph/ http://xxx.lanl.gov/abs/quant- ph/ [8] D. Greve, “QDD: A Quantum Computer Emulation Library,”