The geometry of quantum computation

2008 ◽  
Vol 8 (10) ◽  
pp. 861-899 ◽  
Author(s):  
M.R. Dowling ◽  
M.A. Nielsen
Keyword(s):  
Quantum Computation ◽  
Open Problem ◽  
Complexity Classes ◽  
Real Analysis ◽  
Quantum Oracle ◽  
Quantum Complexity

Whether the class Quantum Merlin Arthur is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a quantum oracle relative to which QMA\neqQMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such trivial containments as BQP\subseteq{ZQEXP}.

scholarly journals On perfect completeness for QMA

2009 ◽  
Vol 9 (1&2) ◽  
pp. 81-89
Author(s):  
S. Aaronson
Keyword(s):  
Open Problem ◽  
Complexity Classes ◽  
Real Analysis ◽  
Quantum Complexity

Whether the class QMA (Quantum Merlin Arthur)\ is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult,\ by using ideas from real analysis to give a "quantum" relative to which QMA \neq QMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP \subseteq ZQEXP.

scholarly journals ON THE POWER QUANTUM COMPUTATION OVER REAL HILBERT SPACES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350001 ◽  
Author(s):  
MATTHEW McKAGUE
Keyword(s):  
Hilbert Space ◽  
Quantum Computation ◽  
Hilbert Spaces ◽  
Quantum Circuit ◽  
Complex Hilbert Space ◽  
Complexity Classes ◽  
Complex Numbers ◽  
Real Numbers ◽  
Single Qubit ◽  
Quantum Complexity

We consider the power of various quantum complexity classes with the restriction that states and operators are defined over a real, rather than complex, Hilbert space. It is well known that a quantum circuit over the complex numbers can be transformed into a quantum circuit over the real numbers with the addition of a single qubit. This implies that BQP retains its power when restricted to using states and operations over the reals. We show that the same is true for QMA (k), QIP (k), QMIP and QSZK.

scholarly journals The Jones polynomial: quantum algorithms and applications in quantum complexity theory

2008 ◽  
Vol 8 (1&2) ◽  
pp. 147-180
Author(s):  
P. Wocjan ◽  
J. Yard
Keyword(s):  
Quantum Computation ◽  
Braid Group ◽  
Quantum Algorithms ◽  
Primitive Root ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Complete Problems ◽  
Primitive Roots ◽  
Quantum Complexity

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

Quantum computing and polynomial equations over Z_2

2005 ◽  
Vol 5 (2) ◽  
pp. 102-112
Author(s):  
C.M. Dawson ◽  
H.L. Haselgrove ◽  
A.P. Hines ◽  
D. Mortimer ◽  
M.A. Nielsen ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Finite Field ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Complexity Classes ◽  
Polynomial Equations ◽  
Computational Power ◽  
Number Of Solutions

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z_2. This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP/P/\#P and BQP/PP.

Method for Organizing Grover's Quantum Oracle

2014 ◽  
Vol 21 (04) ◽  
pp. 1450011
Author(s):  
Hideaki Ito ◽  
Saburou Iida
Keyword(s):  
Quantum Computation ◽  
Knapsack Problem ◽  
Time Complexity ◽  
Quantum Circuit ◽  
Space Complexity ◽  
Complete Problems ◽  
Black Boxes ◽  
Np Complete ◽  
Quantum Oracle ◽  
Toffoli Gates

In a quantum computation, some algorithms use oracles (black boxes) for abstract computational objects. This paper presents an example for organizing Grover's quantum oracle by synthesizing several unitary gates such as CNOT gates, Toffoli gates, and Hadamard gates. As an example, we show a concrete quantum circuit for the knapsack problem, which belongs to the class of NP-complete problems. The time complexity of an oracle for the knapsack problem is estimated to be O(n2), where n is the number of variables. And the same order is obtained for space complexity.

scholarly journals Polynomial-Time Random Oracles and Separating Complexity Classes

2021 ◽  
Vol 13 (1) ◽  
pp. 11-16
Author(s):  
John M. Hitchcock ◽  
Adewale Sekoni ◽  
Hadi Shafei
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Random Oracle ◽  
Complexity Class ◽  
Complexity Classes ◽  
Random Oracles ◽  
Class Separation ◽  
Polynomial Time Hierarchy

Bennett and Gill [1981] showed that P A ≠ NP A ≠ coNP A for a random oracle A , with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that P A ≠ NP A for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If P A ≠ NP A relative to every p-random oracle A , then BPP ≠ EXP. (3) If P A ≠ NP A relative to some p-random oracle A , then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PH A is infinite relative to oracles A that are p-betting-game random. Showing that PH A separates at even its first level would also imply an unrelativized complexity class separation: (4) If NP A ≠ coNP A for a p-betting-game measure 1 class of oracles A , then NP ≠ EXP. (5) If PH A is infinite relative to every p-random oracle A , then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) L A ≠ P A relative to every oracle A that is p-betting-game random.

scholarly journals Counting, fanout and the complexity of quantum ACC

2002 ◽  
Vol 2 (1) ◽  
pp. 35-65
Author(s):  
F. Green ◽  
S. Homer ◽  
C. Moore ◽  
C. Pollett
Keyword(s):  
Complexity Classes ◽  
Constant Depth ◽  
Classical Counterpart ◽  
Classical Class ◽  
Polynomial Size ◽  
Constant Depth Circuits ◽  
Language Classes ◽  
Threshold Circuits ◽  
Quantum Analog ◽  
Quantum Complexity

We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

THE EFFECT OF INKDOTS FOR TWO-DIMENSIONAL AUTOMATA

1995 ◽  
Vol 09 (05) ◽  
pp. 777-796 ◽  
Author(s):  
AKIRA ITO ◽  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI ◽  
YUE WANG
Keyword(s):  
Turing Machine ◽  
Open Problem ◽  
Complexity Classes ◽  
Two Dimensional ◽  
Turing Machines ◽  
Input Tape ◽  
One Dimensional ◽  
Finite State ◽  
Relationship Of ◽  
The Relationship

Recently, related to the open problem of whether deterministic and nondeterministic space (especially lower-level) complexity classes are separated, the inkdot Turing machine was introduced. An inkdot machine is a conventional Turing machine capable of dropping an inkdot on a given input tape for a landmark, but not to pick it up nor further erase it. In this paper, we introduce a finite state version of the inkdot machine as a weak recognizer of the properties of digital pictures, rather than a Turing machine supplied with a one-dimensional working tape. We first investigate the sufficient spaces of three-way Turing machines to simulate two-dimensional inkdot finite automaton, as preliminary results. Next, we investigate the basic properties of two-dimensional inkdot automaton, i.e. the hierarchy based on the number of inkdots and the relationship of two-dimensional inkdot automata to other conventional two-dimensional automata. Finally, we investigate the recognizability of connected pictures of two-dimensional inkdot finite machines.

scholarly journals Uncountable classical and quantum complexity classes

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 111-126
Author(s):  
Maksims Dimitrijevs ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Linear Space ◽  
Time Constant ◽  
Complexity Classes ◽  
Turing Machines ◽  
Small Space ◽  
Logarithmic Space ◽  
Unary Languages ◽  
Bounded Error ◽  
Counter Machines ◽  
Quantum Complexity

It is known that poly-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (A.C. Cem Say and A. Yakaryılmaz, Magic coins are useful for small-space quantum machines. Quant. Inf. Comput. 17 (2017) 1027–1043). In this paper, we investigate more restricted cases for both models to recognize uncountably many languages with bounded error. We show that double logarithmic space is enough for PTMs on unary languages in sweeping reading mode or logarithmic space for one-way head. On unary languages, for quantum models, we obtain middle logarithmic space for counter machines. For binary languages, arbitrary small non-constant space is enough for PTMs even using only counter as memory. For counter machines, when restricted to polynomial time, we can obtain the same result for linear space. For constant-space QTMs, we obtain the result for a restricted sweeping head, known as restarting realtime.

Quantum Computational Complexity Theory and Lower Bounds

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Computational Complexity ◽  
Lower Bounds ◽  
Quantum Computer ◽  
Black Box ◽  
Box Model ◽  
Quantum Computers ◽  
Complexity Classes ◽  
Integer Multiplication ◽  
Classical Computer ◽  
Quantum Complexity

We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.

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