1 2 What is Quantum Computing? Quantum Computing Lecture 1 Aim to use quantum mechanical phenomena that have no classical counterpart for computational purposes. Anuj Dawar Central research tasks include: Building devices — with a specified behaviour. • Designing algorithms — to use the behaviour. Bits and Qubits • Mediating these two are models of computation. 3 4 Bird’s eye view Why look at Quantum Computing? A computer scientist looks at Quantum Computing: The world is quantum • – classical models of computation provide a level of Algorithmic Languages abstraction – discrete state systems Theory/complexity Devices are getting smaller • System Architecture – Moore’s law – Specified Behaviour the only descriptions that work on the very small scale are quantum Physics Exploit quantum phenomena • – using quantum phenomena may allow us to perform Dragons computational tasks that are not otherwise possible/efficient – understand capabilities/resources 5 6 Course Outline Useful Information A total of eight lecturers. Some useful books: 1. Bits and Qubits (this lecture). Nielsen, M.A. and Chuang, I.L. (2000). Quantum Computation • and Quantum Information. Cambridge University Press. 2. Linear Algebra Mermin, N.D. (2007). Quantum Computer Science. CUP. 3. Quantum Mechanics • Kitaev, A.Y., Shen, A.H. and Vyalyi, M.N. (2002). Classical 4. Models of Computation • and Quantum Computation. AMS. 5. Some Applications Hirvensalo, M. (2004). Quantum Computing. Springer. 6. Search Algorithms • Course website: 7. Factorisation http://www.cl.cam.ac.uk/teaching/0910/QuantComp/ 8. Complexity 7 8 Bits Qubits A building block of classical computational devices is a two-state Quantum mechanics tells us that any such system can exist in a system. superposition of states. 0 1 In general, the state of a quantum bit (or qubit for short) is ←→ described by: α 0 + β 1 | i | i Indeed, any system with a finite set of discrete, stable states, with where, α and β are complex numbers, satisfying controlled transitions between them will do. α 2 + β 2 = 1 | | | | 9 10 Qubits Measurement Any attempt to measure the state A qubit may be visu- α 0 + β 1 1 alised as a unit vector | i | i | i β α 0 + β 1 on the plane. results in 0 with probability α 2, and 1 with probability β 2. | i | i | i | | | i | | In general, however, After the measurement, the system is in the measured state! 0 α and β are complex α | i numbers. That is, further measurements will always yield the same value. We can only extract one bit of information from the state of a qubit. 11 12 Measurement Vectors Formally, the state of a qubit is a unit vector in C2—the 1 2-dimensional complex vector space. | i β α 0 + β 1 α 0 + β 1 and α 0 β 1 have | i | i α | i | i | i− | i The vector can be written as the same probabilities for their β measurement 0 α | i α 0 + β 1 | i | i However, they are distinct states α 0 β 1 1 0 which behave differently in terms | i− | i where, 0 = and 1 = . of how they evolve. | i 0 | i 1 φ — a ket, Dirac notation for vectors. | i 13 14 Basis Example Any pair of vectors φ , ψ C2 that are linearly independent 1 | i | i∈ The vector √2 measured in the computational basis gives could serve as a basis. 1 " √2 # either outcome with probability 1/2. α 0 + β 1 = α′ φ + β′ ψ | i | i | i | i Measured in the basis 1 1 The basis is determined by the measurement process or device. √2 , √2 1 1 √2 √−2 Most of the time, we assume a standard (orthonormal) basis 0 | i it gives the first outcome with probability 1. and 1 is given. | i This will be called the computational basis 15 16 Entanglement Entanglement An n-qubit system can exist in any superposition of the 2n basis Compare the two (2-qubit) states: states. 1 1 ( 00 + 01 ) and ( 00 + 11 ) √ | i | i √ | i | i α0 000000 + α1 000001 + + α2n 1 111111 2 2 | i | i ··· − | i 2n 1 2 with i=0− αi = 1 | | If we measure the first qubit in the first case, we see 0 with | i P probability 1 and the state remains unchanged. Sometimes such a state can be decomposed into the states of individual bits In the second case (an EPR pair), measuring the first bit gives 0 1 1 1 | i ( 00 + 01 + 10 + 11 )= ( 0 + 1 ) ( 0 + 1 ) or 1 with equal probability. After this, the second qubit is also 2 | i | i | i | i √2 | i | i ⊗ √2 | i | i | i determined. 1 2 Linear Algebra Quantum Computing Lecture 2 The state space of a quantum system is described in terms of a vector space. Anuj Dawar Vector spaces are the object of study in Linear Algebra. In this lecture we review definitions from linear algebra that we Review of Linear Algebra need in the rest of the course. We are mainly interested in vector spaces over the complex number field – C. We use the Dirac notation—|vi, |φi (read as ket) for vectors. 3 4 Vector Spaces Cn A vector space over C is a set V with α1 Cn . • a commutative, associative addition operation + that has is the vector space of n-tuples of complex numbers: . – 0 0 αn an identity : |vi + = |vi – inverses: |vi +(−|vi)= 0 α1 β1 α1 + β1 . • an operation of multiplication by a scalar α ∈ C such that: with addition . + . = . – α(β|vi)=(αβ)|vi αn βn αn + βn – (α + β)|vi = α|vi + β|vi and α(|ui + |vi)= α|ui + α|vi α1 zα1 – 1|vi = |vi. and scalar multiplication z . = . αn zαn 5 6 Basis Bases for Cn A basis of a vector space V is a minimal collection of vectors 1 0 0 |v1i,..., |vni such that every vector |vi∈ V can be expressed as a 0 1 0 The standard basis for Cn is , ,..., linear combination of these: . . . . |vi = α1|v1i + ··· + αn|vni. 0 0 1 (written |0i,..., |n − 1i). 3 4 2 n—the size of the basis—is uniquely determined by V and is called But other bases are possible: , is a basis for C . 2 −i the dimension of V. " # " # We’ll be interested in orthonormal bases. That is bases of vectors Given a basis, every vector |vi can be represented as an n-tuple of of unit length that are mutually orthogonal. Examples are |0i, |1i scalars. 1 1 and √2 (|0i + |1i), √2 (|0i−|1i). 7 8 Linear Operators Matrices A linear operator A from one vector space V to another W is a Given a choice of bases |v1i,..., |vni and |w1i,..., |wmi, let function such that: m A|vj i = αij|wii A(α|ui + β|vi)= α(A|ui)+ β(A|vi) i=1 X Then, the matrix representation of A is given by the entries αij. If V is of dimension n and W is of dimension m, then the operator A can be represented as an m × n-matrix. Multiplying this matrix by the representation of a vector |vi in the basis |v1i,..., |vni gives the representation of A|vi in the basis |w1i,..., |w i. The matrix representation depends on the choice of bases for V m and W. 9 10 Examples Inner Products 1 1 An inner product on V is an operation that associates to each pair A 45 rotation of the real plane that takes to √2 and ◦ 1 |ui, |vi of vectors a complex number " 0 # " √2 # − 1 0 √2 hu|vi. to 1 is represented, in the standard basis by the " 1 # " √2 # matrix The operation satisfies 1 1 √2 − √2 • hu|αv + βwi = αhu|vi + βhu|wi 1 1 √2 √2 • hu|vi = hv|ui∗ where the ∗ denotes the complex conjugate. • hv|vi≥ 0 (note: hv|vi is a real number) and hv|vi = 0 iff 0 −i The operator does not correspond to a transformation |vi = 0. i 0 of the real plane. 11 12 Inner Product on Cn Norms The standard inner product on Cn is obtained by taking, for The norm of a vector |vi (written |||vi||) is the non-negative, real number: |ui = ui|ii and |vi = vi|ii v v v . i i ||| i|| = h | i X X p A unit vector is a vector with norm 1. hu|vi = ui∗vi i X Two vectors |ui and |vi are orthogonal if hu|vi = 0. Note: hu| is a bra, which together with |vi forms the bra-ket hu|vi. An orthonormal basis for an inner product space V is a basis made up of pairwise orthogonal, unit vectors. the term Hilbert space is also used for an inner product space 13 14 Outer Product Eigenvalues With a pair of vectors |ui∈ U, |vi∈ V we associate a linear An eigenvector of a linear operator A : V → V is a non-zero vector operator |uihv| : V → U, known as the outer product of |ui and |vi. |vi such that A|vi = λ|vi (|uihv|)|v′i = hv|v′i|ui for some complex number λ λ is the eigenvalue corresponding to the eigenvector v. |vihv| is the projection on the one-dimensional space generated by |vi. The eigenvalues of A are obtained as solutions of the characteristic equation: Any linear operator can be expressed as a linear combination of det(A − λI) = 0 outer products: A = Aij|iihj|. Each operator has at least one eigenvalue. ij X 15 16 Diagonal Representation Adjoints A linear operator A is diagonalisable if Associated with any linear operator A is its adjoint A† which satisfies A = λi|viihvi| i hv|Awi = hA†v|wi X where the |vii are an orthonormal set of eigenvectors of A with T corresponding eigenvalues λi.