IPQI-2010-Anu Venugopalan 1 qubits, quantum registers and gates Anu Venugopalan Guru Gobind Singh Indraprastha Univeristy Delhi _______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-2010) Institute of Physics (IOP), Bhubaneswar January 2010

IPQI-2010-Anu Venugopalan 2 The Qubit ______________________________________ Normalization Physical implementations - Photons, electron, spin, nuclear spin ‘Bit’ : fundamental concept of classical computation & info. - 0 or 1 ‘Qubit’ : fundamental concept of quantum computation & info - can be thought of mathematical objects having some specific properties

IPQI-2010-Anu Venugopalan 3 The Qubit- computational basis ______________________________________ General state of a qubit and are complex coefficients that can take any possible values and satisfy the normalization condition: Computational basis State space : C 2 orthonormal basis

IPQI-2010-Anu Venugopalan 4 The Qubit- measurement in the computational basis _________________________________ In general the state of a qubit is a unit vector in a two dimensional complex vector space. Unlike a bit you cannot ‘examine’ a qubit to determine its quantum state

IPQI-2010-Anu Venugopalan 5 The Qubit- measurement in the computational basis _________________________________ - A qubit can exist in a continuum of states between and until it is observed Measuring on the qubit: measurement with prob

IPQI-2010-Anu Venugopalan 6 How much information does a qubit hold? _________________________________ Geometric representation in terms of a Bloch sphere: A point on a unit 3D sphere There are an infinite number of points on the unit sphere, so that in principle one could store a large amount of information But – from a single measurement one obtains only a single bit of information. In the state of a qubit, Nature conceals a great deal of hidden information

IPQI-2010-Anu Venugopalan 7 multiple qubits - quantum registers _________________________________ - More than one qubit….. The state space of a composite physical system is the tensor product of the state spaces of the component systems. Example: for a two qubits, the state space is C 2 C 2 =C 4 computational basis for C 2 : computational basis for C 4 : alternate representation :

IPQI-2010-Anu Venugopalan 8 Two qubit register – computational basis _________________________________ computational basis : The state vector for two qubits: state of qubit 1 state of qubit 2

IPQI-2010-Anu Venugopalan 9 Two qubit register – matrix representation _________________________________ computational basis : state of qubit 1 state of qubit 2

IPQI-2010-Anu Venugopalan 10 Two qubit register – matrix representation _________________________________ The state vector for two qubits:

IPQI-2010-Anu Venugopalan 11 n-qubit register _________________________________ Two-qubit register- state space C 2 ; 4 basis states 4 terms in the superposition for the state vector C 8 :three-qubit register : 2 3 =8 terms in the superposition C 16 :four-qubit register : 2 4 =16 terms in the superposition C n :n-qubit register : 2 n terms in the superposition

IPQI-2010-Anu Venugopalan 12 n-qubit register _________________________________ n-qubit register- 2 n basis states 2 n terms in the superposition for the state vector : a superposition specified by 2 n amplitudes e.g. for n=500 (a quantum register of 500 qubits), the number of terms in the superposition, i.e., 2 500, is larger than the number of atoms in the Universe! A few hundered atoms can store an enormous amount of data - an exponential amount of classical info. in only a polynomial number of qubits because of the superposition.

IPQI-2010-Anu Venugopalan 13 Computing – gates – quantum analogs _________________________________ - Quantum Mechanics as computation Classical computer circuits consist of logic gates. The logic gates perform manipulations of the information, converting it from one form to another. Quantum analogs of logic gates are Unitary operators which can be represented as matrices. Unitary operators (quantum gates ) operate on qubits and quantum registers.

IPQI-2010-Anu Venugopalan 14 Computing –gates – quantum analogs _________________________________ example: single qubit quantum gates: The NOT Gate

IPQI-2010-Anu Venugopalan 15 Computing –gates – quantum analogs _________________________________ example: single qubit quantum gates: The phase flip Gate

IPQI-2010-Anu Venugopalan 16 Computing –gates – quantum analogs _________________________________ example: single qubit quantum gates: The Hadamard Gate This gate is uniquely quantum-mechanical with no classical counterpart

IPQI-2010-Anu Venugopalan 17 Computing –gates – quantum analogs _________________________________ example: single qubit quantum gates: The Hadamard Gate

IPQI-2010-Anu Venugopalan 18 Computing –gates – quantum analogs _________________________________ example: two-qubit quantum gates: The quantum controlled NOT gate The classical C-NOT gate The Quantum C-NOT gate (reversible) control qubit target qubit action

IPQI-2010-Anu Venugopalan 19 Quantum Gates – the C-NOT gate _________________________________ Quantum C-NOT gate is reversible- it corresponds to a Unitary operator, matrix representation

IPQI-2010-Anu Venugopalan 20 Quantum Gates – the swap circuit _________________________________ Three quantum C-NOT gates

IPQI-2010-Anu Venugopalan 21 A quantum circuit for producing Bell states _________________________________ These are very useful states

IPQI-2010-Anu Venugopalan 22 The No-Cloning Theorem _________________________________ Copying/cloning a single classical bit Use a C- NOT gate x 0 x y x xyxy x x  Two bits of the same input x The C-NOT operation Can we have a similar quantum circuit that can clone/copy a qubit?

IPQI-2010-Anu Venugopalan 23 The No-Cloning Theorem _________________________________ A quantum C-NOT gate to clone/copy a qubit? X= Input state Action of C-NOT output state

IPQI-2010-Anu Venugopalan 24 The No-Cloning Theorem _________________________________ Can a quantum C-NOT gate clone/copy a qubit? X= Input state output state If the circuit had cloned the input state x as in the case of the classical circuit, the output state should be XXXX

IPQI-2010-Anu Venugopalan 25 The No-Cloning Theorem _________________________________ X= Input state: output state: If the circuit had cloned the input state x as in the case of the classical circuit, the output state should be XXXX

IPQI-2010-Anu Venugopalan 26 The No-Cloning Theorem _________________________________ Input state: output state: Output state expected of a cloning machine clearly  We have not managed to clone the state

IPQI-2010-Anu Venugopalan 27 The No-Cloning Theorem _________________________________  We have not managed to clone the state if RHS = LHS Cloning happens only if the input state is either or - These are like classical bits!

IPQI-2010-Anu Venugopalan 28 The No-Cloning Theorem _________________________________ Cloning happens only if the input state is either or Only orthogonal states (classical bits) can be cloned It is impossible to clone an unknown quantum state like an input state of the qubit = The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state.

IPQI-2010-Anu Venugopalan 29 The No-Cloning Theorem _________________________________ The no cloning theorem was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields. The theorem follows from the fact that all quantum operations must be unitary linear transformation on the state W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802–803. D. Dieks, Communication by EPR devices, Physics Letters A, vol. 92(6) (1982), pp. 271–272. V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.

IPQI-2010-Anu Venugopalan 30 Some consequences of the no-cloning theorem ___________________________________ The no cloning theorem prevents us from using classical error correction techniques on quantum states. the no cloning theorem is a vital ingredient in quantum cryptography, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key. Fundamentally, the no-cloning theorem protects the uncertainty principle in quantum mechanics More fundamentally, the no cloning theorem prevents superluminal communication via quantum entanglement.