Quantum Computers Algorithms and applications

Simulating classical operations 2/41 Dušan Gajević

Simulating classical operations Is it possible to simulate classical computer operations on a quantum computer? – Pure states correspond to classical bits – To implement any classical computer operation a universal set of gates is needed – (NOT, AND), (NOT, OR), (NAND), (NOR)… Quantum computers are reversible, yet, in any of these sets there is at least one irreversible operation Obviously, the real question is, can a reversible computer simulate an irreversible operation? 3/41 Dušan Gajević

Simulating classical operations – Toffoli gate can be used to implement classical irreversible operations – Is there anything else a reversible circuit might need to simulate a classical computer? NOTANDNAND 4/41 Dušan Gajević

Simulating classical operations – In fact, classical computer contains two more “gates” (elements) that are easily overlooked – Using Toffoli gate FANOUTERASE …but not the actual ERASE Even a type of erase… 5/41 Dušan Gajević

Simulating classical operations – In reversible computing, it is allowed to erase duplicates of information, but it is not allowed to erase the last copy It is possible to simulate any classical irreversible computation using only Toffoli gates, but it requires a use of some extra bits Put another way, Toffoli gate is a universal gate for classical reversible computing 6/41 Dušan Gajević

Computation based on reversible logic can take no energy to compute! Offtopic: Energy dissipation Run forward to get an answer… …copy that answer……undo the computation This will recover all the energy spent except the small amount used for copying an answer 7/41 Dušan Gajević

Offtopic: Energy dissipation – Energy must be dissipated to initialize the system or to make a permanent record of an answer These operations set a new value in a memory register regardless of what the state was, hence, they are irreversible 8/41 Dušan Gajević

Simulating classical operations We saw it’s possible to simulate irreversible operations on a reversible computer Hence, it is possible to simulate classical operations on a quantum computer – But this is not what quantum computers are intended for – it is not much of a benefit… 9/41 Dušan Gajević

Quantum parallelism 10/41 Dušan Gajević

Quantum parallelism A fundamental feature of many quantum algorithms – Suppose a binary function f(x) – Let’s introduce a following unitary transformation single bit in – single bit out XOR (addition modulo 2) 11/41 Dušan Gajević

Quantum parallelism – And use it in a following circuit …state after applying the introduced transformation As if f(x) is evaluated for 2 values of x simultaneously 12/41 Dušan Gajević

– Now a modified binary function f(x) – and a corresponding unitary transformation Quantum parallelism n-bit input single bit output 13/41 Dušan Gajević

Quantum parallelism – in a modified circuit Hadamard transform – applied on n qubits 14/41 Dušan Gajević

Quantum parallelism As if f(x) is evaluated for 2 n values of x simultaneously 15/41 Dušan Gajević

Quantum parallelism To achieve this sort of parallelism in classical computing multiple circuits are needed, while here a single circuit is employed – but, after measurement, only one evaluation of f(x) remains Quantum computing requires more than quantum parallelism alone to work 16/41 Dušan Gajević

“I do not like it, and I am sorry I had anything to do with it” – Erwin Schrödinger 17/41 Dušan Gajević

Deutsch-Jozsa algorithm 18/41 Dušan Gajević

Deutsch-Jozsa algorithm Problem formulation f(x) is allowed to be either: - constant or - balanced A black box… 2 n possible input values output has the same value for all inputs output is: - 0 for half of the function inputs, - 1 for the other half The goal is to determine whether the function f(x) is constant or balanced 19/41 Dušan Gajević

Deutsch-Jozsa algorithm Classical solution: – We are checking the output for every possible input value If we come across a different output – we know the function is balanced Otherwise, when we check at least half plus one possible inputs to be equal (2 n /2 + 1) – we know the function is constant Time complexity of the classical solution is exponential 20/41 Dušan Gajević

Deutsch-Jozsa algorithm classical bits … … … the measurement 21/41 Dušan Gajević

Deutsch-Jozsa algorithm – What makes the resulting state so special? A single amplitude of this state depends on f(x) evaluated for all values of x! 22/41 Dušan Gajević

Deutsch-Jozsa algorithm – Of the special interest is the amplitude of the state 23/41 Dušan Gajević

Deutsch-Jozsa algorithm – there are three different possibilities The function is constant f(x)=0 The function is constant f(x)=1 The function is balanced half f(x)=1 – half f(x)=0 function constant results in 0…0 function balanced results in ≠ 0…0 24/41 Dušan Gajević

Deutsch-Jozsa algorithm Deutsch-Jozsa algorithm: – Compared to the classical solution displayed before, it offers exponential speedup – Unlike most quantum algorithms, it is deterministic – But it is of little practical use – it has no known applications Nevertheless, it suggests quantum computers are capable of solving some problems much more efficiently than conventional computers 25/41 Dušan Gajević

Applications Factorization of large numbers 26/41 Dušan Gajević

The simplest solution is to start from 2 and check for every consequent number whether it’s a factor – Exponential time complexity But even the best known classical algorithms have superpolynomial time complexity Factorization of large numbers prime numbers n digits 27/41 Dušan Gajević

Factorization of large numbers The fact there is no computer today which could efficiently factor numbers is widely used in cryptography for secure data transmission (RSA public key routine) 28/41 Dušan Gajević

Factorization of large numbers Shor’s algorithm – in 1994, Peter Shor published a quantum algorithm for factoring large numbers – Like most quantum algorithms, it is non-deterministic – It has polynomial time complexity – Shor also published a polynomial-time quantum algorithm for discrete logarithms These algorithms sparked a huge amount of interest in quantum computing 29/41 Dušan Gajević

Applications Quantum search 30/41 Dušan Gajević

Quantum search Classically, and on average, how many operations are required for searching an unsorted database with N elements? Grover’s algorithm – in 1996, Lov Grover formulated a quantum algorithm for searching an unstructured database in O(N 1/2 ) time – Another non-deterministic quantum algorithm 31/41 Dušan Gajević

Quantum search The quadratic speedup might not look impressive compared to exponential speedup offered by algorithms based on quantum Fourier transform – But for large search spaces… Quantum search algorithms, also, have a wider range of application than algorithms based on quantum Fourier transformation 32/41 Dušan Gajević

Applications Quantum simulation 33/41 Dušan Gajević

Quantum simulation Simulation of naturally occurring quantum systems – another possible application of quantum computers Why are classical computers so bad at simulating quantum systems? – For a single qubit, a hydrogen atom, two complex coefficients are needed to specify the state of the system – For 2 hydrogen atoms, 4 complex coefficients are needed – For 3 atoms, 8 coefficients – … – For N atoms, 2 N coefficients Number grows exponentially! 34/41 Dušan Gajević

Quantum simulation – For only 30 qubits, more than a billion complex coefficients are needed! Amount of resources needed to simulate a quantum system on a quantum computer grows linearly with the growth of the number of simulated elements – But the extraction of desired information still poses a problem, due to collapse caused by measurement Obviously, quantum systems are really good at spending classical resources 35/41 Dušan Gajević

Technical challenges 36/41 Dušan Gajević

Quantum decoherence There are a lot of technical challenges in building a large-scale quantum computer One of the greatest challenges is minimizing quantum decoherence – Quantum system can get entangled with the environment and evolve as if the environment “measured” the quantum system – Decoherence is irreversible, as it is non-unitary Quantum systems have to be isolated as much as possible from its environment as any unintentional interaction with the outside world can disturb the fragile state of the system 37/41 Dušan Gajević

Advances 38/41 Dušan Gajević

Advances In 2010, company named D-Wave Systems announced D-Wave One, the first commercially available quantum computer system – 128-qubit quantum processor acting as a co-processor, accelerator Intended for solving discrete optimization problems – Front-ended on a network as a standard server – Adiabatic model, unlike gate model explained earlier Quantum annealing – Operating in an extreme environment – Low power consumption - power demand expected to remain constant with scaling up 39/41 Dušan Gajević

Advances In 2013, D-Wave Systems launched 512-qubit D-Wave Two Price? Controversy 40/41 Dušan Gajević

References University of California, Berkeley, Qubits and Quantum Measurement and Entanglement, lecture notes, Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, Colin P. Williams, Explorations in Quantum Computing, Springer, London, Samuel L. Braunstein, Quantum Computation Tutorial, electronic document University of York, York, UK Bernhard Ömer, A Procedural Formalism for Quantum Computing, electronic document, Technical University of Vienna, Vienna, Austria, Artur Ekert, Patrick Hayden, Hitoshi Inamori, Basic Concepts in Quantum Computation, electronic document, Centre for Quantum Computation, University of Oxford, Oxford, UK, Wikipedia, the free encyclopedia, /41 Dušan Gajević