26 research outputs found
DEMONIC programming: a computational language for single-particle equilibrium thermodynamics, and its formal semantics
Maxwell's Demon, 'a being whose faculties are so sharpened that he can follow
every molecule in its course', has been the centre of much debate about its
abilities to violate the second law of thermodynamics. Landauer's hypothesis,
that the Demon must erase its memory and incur a thermodynamic cost, has become
the standard response to Maxwell's dilemma, and its implications for the
thermodynamics of computation reach into many areas of quantum and classical
computing. It remains, however, still a hypothesis. Debate has often centred
around simple toy models of a single particle in a box. Despite their
simplicity, the ability of these systems to accurately represent thermodynamics
(specifically to satisfy the second law) and whether or not they display
Landauer Erasure, has been a matter of ongoing argument. The recent
Norton-Ladyman controversy is one such example.
In this paper we introduce a programming language to describe these simple
thermodynamic processes, and give a formal operational semantics and program
logic as a basis for formal reasoning about thermodynamic systems. We formalise
the basic single-particle operations as statements in the language, and then
show that the second law must be satisfied by any composition of these basic
operations. This is done by finding a computational invariant of the system. We
show, furthermore, that this invariant requires an erasure cost to exist within
the system, equal to kTln2 for a bit of information: Landauer Erasure becomes a
theorem of the formal system. The Norton-Ladyman controversy can therefore be
resolved in a rigorous fashion, and moreover the formalism we introduce gives a
set of reasoning tools for further analysis of Landauer erasure, which are
provably consistent with the second law of thermodynamics.Comment: In Proceedings QPL 2015, arXiv:1511.01181. Dominic Horsman published
previously as Clare Horsma
The ZX calculus is a language for surface code lattice surgery
A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable
Graphical Structures for Design and Verification of Quantum Error Correction
We introduce a high-level graphical framework for designing and analysing
quantum error correcting codes, centred on what we term the coherent parity
check (CPC). The graphical formulation is based on the diagrammatic tools of
the zx-calculus of quantum observables. The resulting framework leads to a
construction for stabilizer codes that allows us to design and verify a broad
range of quantum codes based on classical ones, and that gives a means of
discovering large classes of codes using both analytical and numerical methods.
We focus in particular on the smaller codes that will be the first used by
near-term devices. We show how CSS codes form a subset of CPC codes and, more
generally, how to compute stabilizers for a CPC code. As an explicit example of
this framework, we give a method for turning almost any pair of classical
[n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple
technique for machine search which yields thousands of potential codes, and
demonstrate its operation for distance 3 and 5 codes. Finally, we use the
graphical tools to demonstrate how Clifford computation can be performed within
CPC codes. As our framework gives a new tool for constructing small- to
medium-sized codes with relatively high code rates, it provides a new source
for codes that could be suitable for emerging devices, while its zx-calculus
foundations enable natural integration of error correction with graphical
compiler toolchains. It also provides a powerful framework for reasoning about
all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at
https://doi.org/10.15128/r1bn999672
Communication through coherent control of quantum channels
A completely depolarising quantum channel always outputs a fully mixed state
and thus cannot transmit any information. In a recent Letter [D. Ebler et al.,
Phys. Rev. Lett. 120, 120502 (2018)], it was however shown that if a quantum
state passes through two such channels in a quantum superposition of different
orders - a setup known as the "quantum switch" - then information can
nevertheless be transmitted through the channels. Here, we show that a similar
effect can be obtained when one coherently controls between sending a target
system through one of two identical depolarising channels. Whereas it is
tempting to attribute this effect in the quantum switch to the indefinite
causal order between the channels, causal indefiniteness plays no role in this
new scenario. This raises questions about its role in the corresponding effect
in the quantum switch. We study this new scenario in detail and we see that,
when quantum channels are controlled coherently, information about their
specific implementation is accessible in the output state of the joint
control-target system. This allows two different implementations of what is
usually considered to be the same channel to therefore be differentiated. More
generally, we find that to completely describe the action of a coherently
controlled quantum channel, one needs to specify not only a description of the
channel (e.g., in terms of Kraus operators), but an additional "transformation
matrix" depending on its implementation.Comment: 14 pages, 2 figure
The Natural Science of Computing
As unconventional computing comes of age, we believe a revolution is needed in our view of computer science
SZX-Calculus: Scalable Graphical Quantum Reasoning
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes
Quantum Codes from Classical Graphical Models
We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recently-introduced ‘coherent parity check’ quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes
The Information Content of Systems in General Physical Theories
What kind of object is a quantum state? Is it an object that encodes an
exponentially growing amount of information (in the size of the system) or more
akin to a probability distribution? It turns out that these questions are
sensitive to what we do with the information. For example, Holevo's bound tells
us that n qubits only encode n bits of classical information but for certain
communication complexity tasks there is an exponential separation between
quantum and classical resources. Instead of just contrasting quantum and
classical physics, we can place both within a broad landscape of physical
theories and ask how non-quantum (and non-classical) theories are different
from, or more powerful than quantum theory. For example, in communication
complexity, certain (non-quantum) theories can trivialise all communication
complexity tasks. In recent work [C. M. Lee and M. J. Hoban, Proc. Royal Soc. A
472 (2190), 2016], we showed that the immense power of the information content
of states in general (non-quantum) physical theories is not limited to
communication complexity. We showed that, in general physical theories, states
can be taken as "advice" for computers in these theories and this advice allows
the computers to easily solve any decision problem. Aaronson has highlighted
the close connection between quantum communication complexity and quantum
computations that take quantum advice, and our work gives further indications
that this is a very general connection. In this work, we review the results in
our previous work and discuss the intricate relationship between communication
complexity and computers taking advice for general theories.Comment: In Proceedings PC 2016, arXiv:1606.0651