# Quantum Computation And Quantum Information: 10...

Peter Shor called the text "an excellent book". Lov Grover called it "the bible of the quantum information field". Scott Aaronson said about it, "'Mike and Ike' as it's affectionately called, remains the quantum computing textbook to which all others are compared."[7] David DiVincenzo said, "More than any of the previous attempts, this book has identified the essential foundations of quantum information theory with a clarity that has even, in a few cases, permitted the authors to obtain some original results and point toward new research directions."[8] A review in the November 2001 edition of Foundations of Physics says, "Among the handful of books that have been written on this new subject, the present volume is the most complete and comprehensive."[9]

## Quantum Computation and Quantum Information: 10...

The theory of quantum information and quantum computation. Overview ofclassical information theory, compression of quantum information, transmissionof quantum information through noisy channels, quantum entanglement, quantumcryptography. Overview of classical complexity theory, quantum complexity,efficient quantum algorithms, quantum error-correcting codes, fault-tolerantquantum computation, physical implementations of quantum computation.

Certainly it would be useful to have had a previous course on quantummechanics, though this may not be essential. It would also be useful to knowsomething about (classical) information theory, (classical)coding theory, and (classical) complexity theory, since a central goal ofthe course will be generalize these topics to apply to quantum information.But we will review this material when we get to it, so you don't need to worryif you haven't seen it before. In the discussion of quantum coding, we will usesome rudimentary group theory.

In fact, quantum information -- information storedin the quantum state of a physical system -- has weird properties that contrastsharply with the familiar properties of "classical" information. Anda quantum computer -- a new type of machine that exploits the quantum propertiesof information -- could perform certain types of calculations far moreefficiently than any foreseeable classical computer.

In this course, we will study the properties that distinguish quantuminformation from classical information. And we will see how these propertiescan be exploited in the design of quantum algorithms that solve certainproblems faster than classical algorithms can.

A quantum computer will be much more vulnerable than a conventional digitalcomputer to the effects of noise and of imperfections in the machine.Unavoidable interactions of the device with its surroundings will damage thequantum information that it encodes, a process known as decoherence.Schemes must be developed to overcome this difficulty if quantum computers areever to become practical devices.

The strongest adversary in quantum information science is decoherence, which arises owing to the coupling of a system with its environment1. The induced dissipation tends to destroy and wash out the interesting quantum effects that give rise to the power of quantum computation2, cryptography2 and simulation3. Whereas such a statement is true for many forms of dissipation, we show here that dissipation can also have exactly the opposite effect: it can be a fully fledged resource for universal quantum computation without any coherent dynamics needed to complement it. The coupling to the environment drives the system to a steady state where the outcome of the computation is encoded. In a similar vein, we show that dissipation can be used to engineer a large variety of strongly correlated states in steady state, including all stabilizer codes, matrix product states4, and their generalization to higher dimensions5.

The situation we have in mind is shown in Fig. 1. A quantum system composed of N particles (such as qubits) is organized in space according to a particular geometry (in the figure, a one-dimensional lattice). Neighbouring systems are coupled to some local environments, which are dissipative in nature and tend to drive the system to a steady state. Our idea is to engineer those couplings, so that the environments drive the system to a desired final state. The coupling to the environment will be static, so that the desired state is obtained after some time without having to actively control the system. Note that the role of the environments is to dissipate (or, more precisely, evacuate) the entropy of the system, and by choosing the couplings appropriately we can use this effect to drive our system.

We consider a collection of N quantum particles, locally coupled to a set of environments. The couplings are engineered in such a way that the system reaches the desired state in the long-time limit.

Here, we will concentrate first on DQC, showing how given any quantum circuit one can construct a locally acting master equation for which the steady state is unique, encodes the outcome of the circuit and is reached in polynomial time (with respect to the one corresponding to the circuit). Then we will show how to construct dissipative processes that drive the system to the ground state of any frustration-free Hamiltonian. In the Methods section, we will prove that MPS (ref. 9) and certain kinds of PEPS (ref. 9) can be efficiently prepared using this method, and in Supplementary Information we will give details of the proofs. In this letter we will not consider specific physical set-ups where our ideas can be implemented. Nevertheless, the Methods section will provide a universal way of engineering the master equations required for DQC and DSE, which can be easily adapted to current experiments 13 based on, for example, atoms in optical lattices 14 or trapped ions15. Thus, we expect that our predictions may be experimentally tested in the near future.

In some sense, the present formalism can be seen as a robust way of doing adiabatic quantum computation18 (errors do not accumulate and the path does not have to be engineered carefully) and implementing quantum random walks19, and it might therefore be easier to tackle interesting open questions, such as the quantum probabilistically-checkable-proofs theorem, in this setting20. In addition, it seems that the dissipative way of preparing ground states is more natural than to use adiabatic time evolution, as nature itself prepares them by cooling.

We have investigated the computational power of purely dissipative processes, and proved that it is equivalent to that of the quantum circuit model of quantum computation. We have also shown that dissipative dynamics can be used to create ground states (such as MPS or PEPS) of frustration-free Hamiltonians of strongly correlated quantum spin systems. We believe that these new methods can be experimentally tested using atoms or ions with current set-ups (see the Methods section).

Let us stress that we have been concerned here with a proof-of-principle demonstration that dissipation provides us with an alternative way of carrying out quantum computations or state engineering. We believe, however, that much more efficient and practical schemes can be developed and adapted to specific implementations. We also think that these results open up some interesting questions that deserve further investigation: for example, how the use of fault-tolerant computations can make our scheme more robust, or how one can design translationally invariant completely positive maps that prepare MPS more efficiently, or the importance and generality of the set of commuting Hamiltonians (see the Methods section), which is intimately connected to the fixed points of the renormalization group transformations on PEPS (as it happens with MPS; ref. 25). Furthermore, the model of DQC might well lead to the construction of new quantum algorithms, as, for example, quantum random walks can more easily be formulated within this context. Finally, other ideas related to this work can be easily addressed using the methods introduced; for example, thermal states of commuting Hamiltonians can be engineered using DSE because the Metropolis way of sampling over classical spin configurations can be adopted to the case of commuting operators. Similar techniques could be applied to free fermionic and bosonic systems, and, more generally, it should be possible to devise DSE schemes converging to the ground or thermal states of frustrated Hamiltonians by combining unitary and dissipative dynamics.

Note added. Concurrently with the submission of this paper, refs 26 and 27 appeared in which a similar quantum-reservoir engineering was used to prepare many-body states and non-equilibrium quantum phases.

There will be two in-class exams during the term. In place of a final exam, youare to prepare a term paper, on the order of 15 to 20 pages, to turn in at theend of the course. The topic should have something to do with quantumcomputation or information theory, and must be approved by the instructor. Itis always best to choose something you find interesting or exciting. A onepage proposal will be due at a time to be announced later.

The papers in this volume give readers a broad introduction to the manymathematical research challenges posed by the new and emerging field of quantumcomputation and quantum information. Of particular interest is a long paper byLomonaco and Kauffman discussing mathematical and computational aspects of theso-called hidden subgroup algorithm.

Due to their compactness, robustness, and conciseness, Lindenmayer systems (L-systems) have served as a popular tool for modeling numerous types of systems. Quantum systems, algorithms, and processes maintain a certain regularity to which L-systems are proficient at modeling. In this thesis we explore how one may formulate L-systems in terms of quantum computational areas and the benefits incurred from doing so. This new approach is oriented toward a further extension in the field of L-systems, allowing us to model behavior of quantum computational aspect and more expressively describe quantum processes and phenomena. A particular implementation strategy of how one may invoke an L-system to model these constituents are presented and evaluated. 041b061a72