#### Sponsor

Portland State University. Department of Electrical and Computer Engineering

#### Advisor

Marek Perkowski

#### Date of Award

Spring 5-30-2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.) in Electrical and Computer Engineering

#### Department

Electrical and Computer Engineering

#### Physical Description

1 online resource (xxxvi, 266 pages)

#### Subjects

Quantum computers -- Research, Many-valued logic, Reversible computing, Nearest neighbor analysis (Statistics)

#### DOI

10.15760/etd.1090

#### Abstract

This dissertation describes the development of automated synthesis algorithms that construct reversible quantum circuits for reversible functions with large number of variables. Specifically, the research area is focused on reversible, permutative and fully specified binary and ternary specifications and the applicability of the resulting circuit to the physical limitations of existing quantum technologies.

Automated synthesis of arbitrary reversible specifications is an NP hard, multiobjective optimization problem, where 1) the amount of time and computational resources required to synthesize the specification, 2) the number of primitive quantum gates in the resulting circuit (quantum cost), and 3) the number of ancillary qubits (variables added to hold intermediate calculations) are all minimized while 4) the number of variables is maximized. Some of the existing algorithms in the literature ignored objective 2 by focusing on the synthesis of a single solution without the addition of any ancillary qubits while others attempted to explore every possible solution in the search space in an effort to discover the optimal solution (i.e., sacrificed objective 1 and 4).

Other algorithms resorted to adding a huge number of ancillary qubits (counter to objective 3) in an effort minimize the number of primitive gates (objective 2). In this dissertation, I first introduce the MMDSN algorithm that is capable of synthesizing binary specifications up to 30 variables, does not add any ancillary variables, produces better quantum cost (8-50% improvement) than algorithms which limit their search to a single solution and within a minimal amount of time compared to algorithms which perform exhaustive search (seconds vs. hours). The MMDSN algorithm introduces an innovative method of using the Hasse diagram to construct candidate solutions that are guaranteed to be valid and then selects the solution with the minimal quantum cost out of this subset.

I then introduce the Covered Set Partitions (CSP) algorithm that expands the search space of valid candidate solutions and allows for exploring solutions outside the range of MMDSN. I show a method of subdividing the expansive search landscape into smaller partitions and demonstrate the benefit of focusing on partition sizes that are around half of the number of variables (15% to 25% improvements, over MMDSN, for functions less than 12 variables, and more than 1000% improvement for functions with 12 and 13 variables). For a function of n variables, the CSP algorithm, theoretically, requires n times more to synthesize; however, by focusing on the middle k (k by MMDSN which typically yields lower quantum cost. I also show that using a Tabu search for selecting the next set of candidate from the CSP subset results in discovering solutions with even lower quantum costs (up to 10% improvement over CSP with random selection).

In Chapters 9 and 10 I question the predominant methods of measuring quantum cost and its applicability to physical implementation of quantum gates and circuits. I counter the prevailing literature by introducing a new standard for measuring the performance of quantum synthesis algorithms by enforcing the Linear Nearest Neighbor Model (LNNM) constraint, which is imposed by the today's leading implementations of quantum technology. In addition to enforcing physical constraints, the new LNNM quantum cost (LNNQC) allows for a level comparison amongst all methods of synthesis; specifically, methods which add a large number of ancillary variables to ones that add no additional variables. I show that, when LNNM is enforced, the quantum cost for methods that add a large number of ancillary qubits increases significantly (up to 1200%).

I also extend the Hasse based method to the ternary and I demonstrate synthesis of specifications of up to 9 ternary variables (compared to 3 ternary variables that existed in the literature). I introduce the concept of ternary precedence order and its implication on the construction of the Hasse diagram and the construction of valid candidate solutions. I also provide a case study comparing the performance of ternary logic synthesis of large functions using both a CUDA graphic processor with 1024 cores and an Intel i7 processor with 8 cores. In the process of exploring large ternary functions I introduce, to the literature, eight families of ternary benchmark functions along with a Multiple Valued file specification (the Extended Quantum Specification XQS). I also introduce a new composite quantum gate, the multiple valued Swivel gate, which swaps the information of qubits around a centrally located pivot point.

In summary, my research objectives are as follows:

* Explore and create automated synthesis algorithms for reversible circuits both in binary and ternary logic for large number of variables.

* Study the impact of enforcing Linear Nearest Neighbor Model (LNNM) constraint for every interaction between qubits for reversible binary specifications.

* Advocate for a revised metric for measuring the cost of a quantum circuit in concordance with LNNM, where, on one hand, such a metric would provide a way for balanced comparison between the various flavors of algorithms, and on the other hand, represents a realistic cost of a quantum circuit with respect to an ion trap implementation.

* Establish an open source repository for sharing the results, software code and publications with the scientific community. With the dwindling expectations for a new lifeline on silicon-based technologies, quantum computations have the potential of becoming the future workhorse of computations. Similar to the automated CAD tools of classical logic, my work lays the foundation for creating automated tools for constructing quantum circuits from reversible specifications.

#### Persistent Identifier

http://archives.pdx.edu/ds/psu/9876

#### Recommended Citation

Hawash, Maher Mofeid, "Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model" (2013). *Dissertations and Theses.* Paper 1090.

https://pdxscholar.library.pdx.edu/open_access_etds/1090

10.15760/etd.1090