scholarly journals A linear-optical proof that the permanent is # P -hard

2011 ◽  
Vol 467 (2136) ◽  
pp. 3393-3405 ◽  
Author(s):  
Scott Aaronson
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Universality Theorem ◽  
Linear Optical ◽  
Intuitive Proof

One of the crown jewels of complexity theory is Valiant's theorem that computing the permanent of an n × n matrix is # P -hard. Here we show that, by using the model of linear-optical quantum computing —and in particular, a universality theorem owing to Knill, Laflamme and Milburn—one can give a different and arguably more intuitive proof of this theorem.

scholarly journals High-fidelity linear optical quantum computing with polarization encoding

2006 ◽  
Vol 73 (1) ◽  
Author(s):  
Federico M. Spedalieri ◽  
Hwang Lee ◽  
Jonathan P. Dowling
Keyword(s):  
Quantum Computing ◽  
High Fidelity ◽  
Polarization Encoding ◽  
Linear Optical

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

Quantum Algorithmic Complexities and Entropy

2009 ◽  
Vol 16 (01) ◽  
pp. 1-28 ◽  
Author(s):  
Fabio Benatti
Keyword(s):  
Quantum Computing ◽  
Quantum Information ◽  
Complexity Theory ◽  
Information Sources ◽  
Algorithmic Complexity ◽  
Entropy Rate ◽  
Von Neumann Entropy ◽  
Von Neumann

We review the basics of classical algorithmic complexity theory and two of its quantum extensions that have been prompted by the foreseeable existence of quantum computing devices. In particular, we will examine the relations between these extensions and the von Neumann entropy rate of generic quantum information sources of ergodic type.

scholarly journals QUANTUM DISCORD AND QUANTUM COMPUTING — AN APPRAISAL

2011 ◽  
Vol 09 (07n08) ◽  
pp. 1787-1805 ◽  
Author(s):  
ANIMESH DATTA ◽  
ANIL SHAJI
Keyword(s):  
Quantum Computing ◽  
Computational Complexity ◽  
Complexity Theory ◽  
Quantum Discord ◽  
Primary Motivation ◽  
Classical Simulation ◽  
Classical Correlations ◽  
Quantum And Classical Correlations ◽  
Theory Lead

We discuss models of computing that are beyond classical. The primary motivation is to unearth the cause of non-classical advantages in computation. Completeness results from computational complexity theory lead to the identification of very disparate problems, and offer a kaleidoscopic view into the realm of quantum enhancements in computation. Emphasis is placed on the "power of one qubit" model, and the boundary between quantum and classical correlations as delineated by quantum discord. A recent result by Eastin on the role of this boundary in the efficient classical simulation of quantum computation is discussed. Perceived drawbacks in the interpretation of quantum discord as a relevant certificate of quantum enhancements are addressed.

scholarly journals Towards a Source of Multi-Photon Entangled States for Linear Optical Quantum Computing

2018 ◽  
Author(s):  
B. Villa ◽  
J. Lee ◽  
A. Bennett ◽  
M. Stevenson ◽  
D. Ellis ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Entangled States ◽  
Linear Optical

scholarly journals Fine-grained quantum computational supremacy

2019 ◽  
Vol 19 (13&14) ◽  
pp. 1089-1115
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Probability Distributions ◽  
Fine Grained ◽  
Multiplicative Error ◽  
Universal Quantum ◽  
The One ◽  
Output Probability ◽  
Computing Models

(pp1089-1115) Tomoyuki Morimae and Suguru Tamaki doi: https://doi.org/10.26421/QIC19.13-14-2 Abstracts: Output probability distributions of several sub-universal quantum computing models cannot be classically efficiently sampled unless some unlikely consequences occur in classical complexity theory, such as the collapse of the polynomial-time hierarchy. These results, so called quantum supremacy, however, do not rule out possibilities of super-polynomial-time classical simulations. In this paper, we study ``fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations. First, we focus on two sub-universal models, namely, the one-clean-qubit model (or the DQC1 model) and the HC1Q model. Assuming certain conjectures in fine-grained complexity theory, we show that for any a>0 output probability distributions of these models cannot be classically sampled within a constant multiplicative error and in 2^{(1-a)N+o(N)} time, where N is the number of qubits. Next, we consider universal quantum computing. For example, we consider quantum computing over Clifford and T gates, and show that under another fine-grained complexity conjecture, output probability distributions of Clifford-T quantum computing cannot be classically sampled in 2^{o(t)} time within a constant multiplicative error, where t is the number of T gates.

scholarly journals Fault tolerance in parity-state linear optical quantum computing

2010 ◽  
Vol 82 (2) ◽  
Author(s):  
A. J. F. Hayes ◽  
H. L. Haselgrove ◽  
Alexei Gilchrist ◽  
T. C. Ralph
Keyword(s):  
Quantum Computing ◽  
Fault Tolerance ◽  
Parity State ◽  
Linear Optical

scholarly journals Linear Optical Quantum Computing in a Single Spatial Mode

2013 ◽  
Vol 111 (15) ◽  
Author(s):  
Peter C. Humphreys ◽  
Benjamin J. Metcalf ◽  
Justin B. Spring ◽  
Merritt Moore ◽  
Xian-Min Jin ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Spatial Mode ◽  
Linear Optical
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