Quantum Complexity Classes By: Larisse D. Voufo On: November 28 th, 2006

Introduction 1982 (Trend toward miniaturization and microcircuitry), Paul Benioff & Richard Feynman: Quantum Systems could perform computation. 1985, David Deutch. Quantum Computer  Turing Machine  possibility of new Complexity of algorithms Later On, Universality of Quantum Circuits  Machine independent notion of quantum complexity.

Key quantum property for quantum complexity studies: Randomness of quantum measurement process  Algorithm performed by a quantum computer is probabilistic. (== multiple runs, different results)

Probabilistic Computation vs. Quantum Computation. Nondeterministic Computation (NC) = tree of configurations of NTM Probabilistic Computation = NC where probabilities edges and nodes.  Rules of Classical Probability. Quantum Computation = NC where amplitudes edges and nodes.  Rules of Quantum Probability.

From Classical Complexity classes … P – “easy”: languages decided by polynomial-time TMs NP: languages decided by polynomial-time NTMs.  Guess an answer, verify in polynomial time. Is answer YES? NP-hard: Every hard problem can be polynomially reduced to a problem in this class. NP-complete:  NPC = NP-hard  NP NP-hard  P  {} => P = NP  NP-hard  P  {} => P = NP

From Classical Complexity classes … NPI: Problems in NP of intermediate difficulty  NPI = NP – P – NPC = NP – P – NP-hard Co-NP: Like NP, but Answer is NO (counter-example based)  NP  Co-NP  No proof for: P  NP.

From Classical Complexity classes … AWPP: languages decided by Almost-Wide Probabilistic Polynomial-time NTMs PP: languages decided by polynomial-time NTMs where the majority of paths gives the correct answer. P #P : functions that count the number of accepting paths through an NP machine. P  NP  AWPP  PP  P #P.  P  NP  AWPP  PP  P #P.

From Classical Complexity classes … IP: Problems solvable by an Interactive Proof System. MA: languages decided by a bounded-error probabilistic Merlin-Arthur protocol. BPP: Bounded-error Probabilistic Polynomial Time. “Problems that admit a probabilistic circuit family of polynomial size that always gives the right answer with prob > ½ +  ”. PSPACE: DPs that can be solved in polynomial-space, but may require exponential time.

… to Quantum Complexity Classes: BQP: Bounded-error Quantum Polynomial Time. “DPs that can be solved, with high probability, by polynomial-size quantum circuits”. EQP (QP): Exact version of BQP

… to Quantum Complexity Classes:  P  BPP  BQP  PSPACE  IP = PSPACE  NP  MA  BPP  MA  IP  BQP  P #P  PSPACE  No firm proof for: BPP  BQP (in general)  If P = PSPACE, then P = AWPP “relative to oracle”  NP = AWPP “relative to oracle”  NP  PSPACE (checking if C(x (n), y (n) ) = 1 for each y (m) )  NP  BQP ?

… to Quantum Complexicity Classes: BQNP ( = QMA) QMA-complete QIP  EQP  BQP  QMA  QIP

BPP Interactive Proof System: IP Polynomial Number of Messages ?, r, … Proof (x  L)

Deterministic Polynomial- time TM Merlin-Arthur Protocol: NP Constant Number of Messages ?, r, …

Merlin-Arthur Protocol: MA BPP Constant Number of Messages ?, r, …

Merlin-Arthur Protocol: QMA(C) QMA-Completeness: ground state energy problem: (5-local hamiltonian). BQP Constant Number of Messages ?, r, …

Merlin-Arthur Protocol: QIP Q- Polynomial Number of Messages BQP ?, r, … Proof (x  L)

A model for quantum circuits: Facts: Quantum gate: unitary transformation  reversible gate. Classical Reversible Computer = special case of Quantum Computer. x (n)  y (n) = f(x (n) ) U: |x i >  |y i > |00…0>  Deterministic final measurement

3 Issues with this model: 1.Universality Complete Model There exists no transformation in U(2 n ) that we cannot reach. Simulation of a Q-computer using another Q-computer  complexity classes do not depend on the details of the hardware. 2.Simulating a quantum computer on a classical computer: Better characterize the resources needed. A Classical Computer can still simulate a Q-Computer, despite a polynomial limit on memory space available.

3 Issues with this model: 3. Accuracy == growth of error in measurement as the quantum circuit size increases. NO Polynomial-size circuit family (hard problems) w/ gates of exponential accuracy. An idealized T-gate q-circuit (acceptable accuracy): Error Prob / gate  1/T. Quantum Algorithm w/ prob > ½ +  (in the ideal case)  Gates w/ accuracy T  < O(  ). BQP can really solve hard problems linear improvement of the accuracy of the gates (computation size T).

More on Relationships between Complexity classes  P  BPP  BQP  AWPP  PP  PSPACE. Bernstein and Vazirani: BQP  PSPACE Adelman, Demarrais and Huang: BQP  PP Fortnow and Rogers: BQP  AWPP

Other Complexity Classes Vary from one literature to another… UP, QPSV, NPSV, UPSV, etc…  Elham Kashefi’s PhD thesis (Imperial College London) NQP, C = P, coC = P, etc…  Tarsem S. Purewal Jr (University of Georgia)

Analyzing Quantum Algorithm Performances Over Classical Ones: 1. Non-exponential speedup: Eg: Grover’s Quantum Speed-up of the Search of an unsorted database. 2. “Relativized” Exponential Speed-up  Oracles  BPP  BQP “relative to oracle”. Eg: Simon’s exponential quantum speedup for finding the period of 2 to 1 function. Deutch’s algorithm. 3. Exponential Speed-up for “apparently” hard problems Eg: Shor’s factoring algorithm.

References: Tarsem S. Purewal Jr. “Revisiting a Limit on Efficient Quantum Computation”. ACM Southeastern Conference 2006, Melbourne, FL. March 10, Slides at John Preskill. “Lecture Notes on Physics 229: Quantum Information and Computation”. Sept California Institute of Technology. Tarsem S. Purewal Jr. “Nondeterministic Quantum Query Complexity”. Combinatorics, Algorithms, and Theory Seminar (CATS), University of Georgia, Athens, GA. May 1, Tarsem S. Purewal Jr. “5-local Hamiltonian is QMA-Complete”. Quantum Computing Journal Club, University of Georgia, Athens, GA. June 6, Prof. Tony Hey. “Quantum Computing: an introduction”. Quantum Technology Center. Artur Ekert, Patrick Hayden, Hitoshi Inamori. “Basic concepts in quantum computation”. converted to wiki format by Burgarth 22:37, 8 Jun 2005 (BST). Qbit.com. “Introduction to Quantum Theory”. Elham Kashefi. “Complexity Analysis and Semantics for Quantum Computation”. November 26, Tarsem S. Purewal Jr. Lance Fortnow. “ One Complexity Theorist's View of Quantum Computing ”.

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