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Quantum Computers The basics Introduction Dušan Gajević 2/47 Introduction • Quantum computers use quantum-mechanical phenomena to represent and process data • Quantum mechanics can be described with three basic postulates – The superposition principle - tells us what states are possible in a quantum system – The measurement principle - tells us how much information about the state we can access – Unitary evolution - tells us how quantum system is allowed to evolve from one state to another Dušan Gajević 3/47 Introduction • Atomic orbitals - an example of quantum mechanics Electrons, within an atom, exist in quantized energy levels (orbits) Limiting the total energy… ...limits the electron to k different levels A hydrogen atom – only one electron This atom might be used to store a number between 0 and k-1 Dušan Gajević 4/47 The superposition principle Dušan Gajević 5/47 The superposition principle • The superposition principle states that if a quantum system can be in one of k states, it can also be placed in a linear superposition of these states with complex coefficients • Ways to think about superposition – Electron cannot decide in which state it is – Electron is in more than one state simultaneously Dušan Gajević 6/47 The superposition principle • State of a system with k energy levels “pure” states “ket psi” Bra-ket (Dirac) notation amplitudes Reminder: Dušan Gajević 7/47 The superposition principle • A system with 3 energy levels – examples of valid states Dušan Gajević 8/47 “Very interesting theory – it makes no sense at all” – Groucho Marx Dušan Gajević 9/47 The measurement principle Dušan Gajević 10/47 The measurement principle • The measurement principle says that measurement on the k state system yields only one of at most k possible outcomes and alters the state to be exactly the outcome of the measurement Dušan Gajević 11/47 The measurement principle • It is said that quantum state collapses to a classical state as a result of the measurement Dušan Gajević 12/47 The measurement principle If we try to measure this state... …the system will end up in this state… …and we will also get it as a result of the measurement The probability of a system collapsing to this state is given with Dušan Gajević 13/47 The measurement principle • This means: – We can tell the state we will read only with a certain probability – Repeating the measurement will always yield the same result we got this first time – Amplitudes are lost as soon as the measurement is made, so amplitudes cannot be measured Dušan Gajević 14/47 The measurement principle • Probability of a system collapsing to a state j is given with Does the equation appear more natural now? – One might ask, if amplitudes come down to probabilities when the state is measured, why use complex amplitudes in the first place? • Answer to this will be given later, when we see how system is allowed to evolve from one state to another Dušan Gajević 15/47 “God does not play dice” – Albert Einstein “Don’t tell God what to do” – Niels Bohr Dušan Gajević 16/47 Qubit Dušan Gajević 17/47 Qubit • Isolating two individual levels in our hydrogen atom and the qubit (quantum bit) is born Dušan Gajević 18/47 Qubit • Qubit state • The measurement collapses the qubit state to a classical bit Dušan Gajević 19/47 Vector reprezentation Dušan Gajević 20/47 Vector representation • Pure states of a qubit can be interpreted as orthonormal unit vectors in a 2 dimensional Hilbert space – Hilbert space – N dimensional complex vector space Reminder: Another way to write a vector – as a column matrix Dušan Gajević 21/47 Vector representation • Column vectors (matrices) qubit state pure states a little bit of math Reminder: Scalar multiplication Reminder: Adding matrices Dušan Gajević 22/47 Vector representation • System with k energy levels represented as a vector in k dimensional Hilbert space system state pure states Dušan Gajević 23/47 Entanglement Dušan Gajević 24/47 Entanglement • Let’s consider a system of two qubits – two hydrogen atoms, each with one electron and two "pure" states Dušan Gajević 25/47 Entanglement • By the superposition principle, the quantum state of these two atoms can be any linear combination of the four classical states Does this look familiar? – Vector representation Dušan Gajević 26/47 Entanglement • Let’s consider the separate states of two qubits, A and B – Interpreting qubits as vectors, their joint state can be calculated as their cross (tensor) product Reminder: Tensor product Dušan Gajević 27/47 Entanglement – Cross product in Dirac notation is often written in a bit different manner • The joint state of A and B in Dirac notation Dušan Gajević 28/47 Entanglement Let’s try to decompose to separate states of two qubits all four have to be non-zero at least one has to be zero It’s impossible! at least one has to be zero Dušan Gajević 29/47 Entanglement • States like the one from the previous example are called entangled states and the displayed phenomenon is called the entanglement – When qubits are entangled, state of each qubit cannot be determined separately, they act as a single quantum system – What will happen if we try to measure only a single qubit of an entangled quantum system? Dušan Gajević 30/47 Entanglement Let’s take a look at the same example once again amplitudes probabilities Dušan Gajević 31/47 Entanglement measuring the first qubit value of the first qubit measuring the second qubit value of the second qubit This remains true no matter how large the distance between qubits is! Dušan Gajević 32/47 “Spooky action at a distance” – Albert Einstein Dušan Gajević 33/47 Unitary evolution Dušan Gajević 34/47 Unitary evolution • Unitary evolution means that transformation of the quantum system state does not change the state vector length – Geometrically, unitary transformation is a rigid body rotation of the Hilbert space Dušan Gajević 35/47 Unitary evolution • It comes down to mapping the old orthonormal basis states to new ones – These new states can be described as superpositions of the old ones Dušan Gajević 36/47 Unitary evolution • Unitary transformation of a single qubit – Dirac notation Replace the old basis states… …with new ones – Matrix representation Multiply unitary matrix… …with the old state vector Dušan Gajević 37/47 Unitary evolution • Example of calculus using Dirac notation Qubit is in the state… …applying following (Hadamard) transformation… …results in the state Dušan Gajević 38/47 Unitary evolution • Example of calculus using matrix representation Reminder: matrix multiplication Qubit is in the state… …applying Hadamard transformation… …results in the state Dušan Gajević 39/47 Unitary matrices • Unitary matrices satisfy the condition Conjugate-transpose of U “U-dagger” Inverse of U Reminder: Conjugate-transpose matrix Reminder: Inverse matrix Reminder: Complex conjugate Identity matrix Reminder: Dušan Gajević 40/47 Reversibility Dušan Gajević 41/47 Reversibility • Reversibility is an important property of unitary transformation as a function – knowing the output it is always possible to determine input – What makes an operation reversible? output • AND circuit INPUT OUTPUT A 0 0 1 1 • NOT circuit B 0 1 0 1 A and B 0 0 0 1 INPUT OUTPUT A not A 0 1 1 0 Dušan Gajević 1 0 input A=1 B=1 ? irreversible output input 1 A=0 0 A=1 reversible 42/47 Reversibility – Reversible operation has to be one-to-one – different inputs have to give different outputs and vice-versa • Consequently, reversible operations have the same number of inputs and outputs • Are classical computers reversible? Dušan Gajević 43/47 Reversibility – Similar to AND circuit applies to OR, NAND and NOR, the usual building blocks of classical computers • Hence, in general, classical computers are not reversible Dušan Gajević 44/47 Offtopic: Landauer’s principle • Again, an irreversible operation – NAND circuit INPUT A B 0 0 0 1 1 0 1 1 OUTPUT A nand B 1 1 1 0 We say information is “erased” every time output of NAND is 1 Whenever output of NAND is 1 – input cannot be determined Dušan Gajević 45/47 Offtopic: Landauer’s principle • Landauer’s principle says that energy must be dissipated when information is erased, in the amount Boltzman's constant Absolute temperature – Even if all other energy loss mechanisms are eliminated irreversible operations still dissipate energy • Reversible operations do not erase any information when they are applied Dušan Gajević 46/47 References • • • • • • • University of California, Berkeley, Qubits and Quantum Measurement and Entanglement, lecture notes, http://www-inst.eecs.berkeley.edu/~cs191/sp12/ Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2010. Colin P. Williams, Explorations in Quantum Computing, Springer, London, 2011. 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, 1998. Artur Ekert, Patrick Hayden, Hitoshi Inamori, Basic Concepts in Quantum Computation, electronic document, Centre for Quantum Computation, University of Oxford, Oxford, UK, 2008. Wikipedia, the free encyclopedia, 2014. Dušan Gajević 47/47