LIMDD A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States

Efficient methods for the representation of relevant quantum states and quantum operations are crucial for the simulation and optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing interesting aspects of quantum systems, but their limits are not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, a well-studied method for simulating quantum circuits in the tractable regime. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for a subset of stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that the set of quantum states represented by poly-sized LIMDDs strictly contains the union of stabilizer states and other decision diagram variants. We also provide evidence that LIMDD-based simulation is capable of efficiently simulating some circuits for which both stabilizer-based and other DD-based methods require exponential time. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing. 2012 ACM Subject Classification Mathematics of computing → Decision diagrams

[1]  H. Briegel,et al.  Persistent entanglement in arrays of interacting particles. , 2000, Physical review letters.

[2]  Igor L. Markov,et al.  Efficient Inner-product Algorithm for Stabilizer States , 2012, ArXiv.

[3]  Saburo Muroga,et al.  Binary Decision Diagrams , 2000, The VLSI Handbook.

[4]  David Gosset,et al.  Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates. , 2016, Physical review letters.

[5]  Masahiro Fujita,et al.  Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping , 1993, 30th ACM/IEEE Design Automation Conference.

[6]  Laura Mančinska,et al.  Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask) , 2012, 1210.4583.

[7]  D. Michael Miller,et al.  QMDD: A Decision Diagram Structure for Reversible and Quantum Circuits , 2006, 36th International Symposium on Multiple-Valued Logic (ISMVL'06).

[8]  Renaud Vilmart,et al.  Quantum Multiple-Valued Decision Diagrams in Graphical Calculi , 2021, MFCS.

[9]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[10]  Kristan Temme,et al.  Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters , 2020, Quantum.

[11]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[12]  Kenneth L. McMillan,et al.  Symbolic model checking: an approach to the state explosion problem , 1992 .

[13]  Pierre Marquis,et al.  A Knowledge Compilation Map , 2002, J. Artif. Intell. Res..

[14]  John P. Hayes,et al.  Quantum Circuit Simulation , 2009 .

[15]  Robert Wille,et al.  One-Pass Design of Reversible Circuits: Combining Embedding and Synthesis for Reversible Logic , 2018, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[16]  Mark Howard,et al.  Simulation of quantum circuits by low-rank stabilizer decompositions , 2018, Quantum.

[17]  R. Jozsa,et al.  Computational power of matchgates with supplementary resources , 2020, Physical Review A.

[18]  Eugene M. Luks,et al.  Some Algorithms for Nilpotent Permutation Groups , 1997, J. Symb. Comput..

[19]  Scott Aaronson,et al.  Improved Simulation of Stabilizer Circuits , 2004, ArXiv.

[20]  Martin B. Plenio,et al.  Entanglement on mixed stabilizer states: normal forms and reduction procedures , 2005, quant-ph/0505036.

[21]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

[22]  Enrico Macii,et al.  Algebraic decision diagrams and their applications , 1993, Proceedings of 1993 International Conference on Computer Aided Design (ICCAD).

[23]  Sagar Chaki,et al.  BDD-Based Symbolic Model Checking , 2018, Handbook of Model Checking.

[24]  Scott Sanner,et al.  Affine Algebraic Decision Diagrams (AADDs) and their Application to Structured Probabilistic Inference , 2005, IJCAI.

[25]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[26]  Randal E. Bryant,et al.  Efficient implementation of a BDD package , 1991, DAC '90.

[27]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[28]  R. Jozsa,et al.  Matchgates and classical simulation of quantum circuits , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Randal E. Bryant,et al.  Verification of Arithmetic Circuits with Binary Moment Diagrams , 1995, 32nd Design Automation Conference.

[30]  Roman Orus,et al.  A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States , 2013, 1306.2164.

[31]  Padraic Calpin,et al.  Exploring quantum computation through the lens of classical simulation , 2020 .

[32]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[33]  B. De Moor,et al.  Local unitary versus local Clifford equivalence of stabilizer states , 2005 .

[34]  Peter Love,et al.  Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits , 2018, Physical Review A.

[35]  I. Wegener.  Branching Programs and Binary Deci-sion Diagrams-Theory and Applications , 1987 .

[36]  John P. Hayes,et al.  High-performance QuIDD-based simulation of quantum circuits , 2004, Proceedings Design, Automation and Test in Europe Conference and Exhibition.

[37]  Yuan Feng,et al.  Approximate Equivalence Checking of Noisy Quantum Circuits , 2021, 2021 58th ACM/IEEE Design Automation Conference (DAC).

[38]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

[39]  Masahiro Fujita,et al.  Multi-Terminal Binary Decision Diagrams: An Efficient Data Structure for Matrix Representation , 1997, Formal Methods Syst. Des..

[40]  Mitchell A. Thornton,et al.  On the Skipped Variables of Quantum Multiple-Valued Decision Diagrams , 2011, 2011 41st IEEE International Symposium on Multiple-Valued Logic.

[41]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[42]  D. Rose,et al.  Generalized nested dissection , 1977 .

[43]  David P. DiVincenzo,et al.  Classical simulation of noninteracting-fermion quantum circuits , 2001, ArXiv.

[44]  Richard Rudell.  Dynamic variable ordering for ordered binary decision diagrams , 1993, ICCAD.

[45]  John P. Hayes,et al.  Improving Gate-Level Simulation of Quantum Circuits , 2003, Quantum Inf. Process..

[46]  H. Andersen.  An Introduction to Binary Decision Diagrams , 1997 .

[47]  J. Eisert,et al.  Entanglement in Graph States and its Applications , 2006, quant-ph/0602096.

[48]  Robert Wille,et al.  Advanced Simulation of Quantum Computations , 2017, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[49]  J. Smolin,et al.  Trading Classical and Quantum Computational Resources , 2015, 1506.01396.

[50]  M. Ying,et al.  A Tensor Network based Decision Diagram for Representation of Quantum Circuits , 2020, ACM Trans. Design Autom. Electr. Syst..

[51]  Pierre Marquis,et al.  A Knowledge Compilation Map for Ordered Real-Valued Decision Diagrams , 2014, AAAI.

[52]  Lucas Kocia,et al.  Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits , 2018, Quantum.

[53]  Juraj Hromkovic,et al.  On multi-partition communication complexity , 2004, Inf. Comput..