Quantum information and quantum computation Vorlesung summer semester 2020 Technische Universit¨atBerlin Dr. Gernot Schaller November 9, 2020 2 Contents 1 A brief intro to classical computation9 1.1 The circuit model of classical computation . .9 1.2 Example: Binary addition . 11 1.3 Going quantum? . 12 2 Qubit(s) 13 2.1 One qubit . 13 2.2 Qubit control . 15 2.3 Single qubit gates . 15 2.4 Multiple qubits . 18 2.5 Quantum applications of two qubits . 22 2.5.1 Bell inequalities . 22 2.5.2 Superdense coding . 24 2.5.3 No-cloning theorem . 25 2.5.4 Quantum teleportation . 25 2.6 The partial trace . 27 3 The circuit model of quantum computation 29 3.1 Deutsch and Deutsch-Jozsa problems . 29 3.2 Backward compatibility . 33 3.3 Controlled operations . 34 3.4 Universality of quantum computation . 35 3.4.1 Decomposition of d-level unitaries into 2-level unitary matrices . 36 3.4.2 Decomposition of 2-level unitaries into single qubit and CNOT . 38 3.4.3 Approximating single-qubit and CNOT by discrete gates . 39 3.5 Quantum Fourier Transform . 41 3.6 Quantum search . 45 3.7 Control errors . 49 4 Open quantum systems 51 4.1 An exactly solvable model of decoherence . 51 4.2 Lindblad master equation: Microscopic derivation . 53 4.3 Lindblad master equation: General properties . 60 5 Adiabatic quantum computation 63 5.1 The adiabatic theorem for closed systems . 64 5.2 Adiabatic qubit control . 65 3 4 CONTENTS 5.3 Adiabatic control in presence of thermal dissipation . 67 5.4 The adiabatic Grover search . 68 5.5 Adiabatic approaches to an NP-complete problem . 71 5.6 An adiabatic adder . 74 5.7 An adiabatic multiplier . 76 5.8 The 1d quantum Ising model in a transverse field . 77 5.8.1 Exact Diagonalization . 78 5.8.2 Adiabatic criterion . 85 5.8.3 Non-straight interpolation . 86 6 Information measures 91 6.1 Quantum operations . 91 6.2 Comparing density matrices . 92 6.2.1 Trace Distance . 92 6.2.2 Fidelity . 96 6.2.3 Quantum relative entropy . 97 6.3 Entanglement . 99 6.3.1 Entanglement entropy . 99 6.3.2 Examples . 101 6.3.3 Entanglement of mixed states . 105 6.3.4 Concurrence . 106 6.3.5 Examples . 108 7 Quantum Thermodynamics 115 7.1 Nonequilibrium thermodynamics . 115 7.1.1 Example: Two coupled qubits . 119 7.1.2 Numeric example: Ising model . 120 7.1.3 Example: Transient entropy production for pure-dephasing . 121 7.1.4 Use of Spohn's inequality . 123 7.2 Quantum Otto cycle . 125 7.2.1 Modeling of closed (unitary) strokes . 126 7.2.2 Modeling of open (dissipative) strokes . 127 7.2.3 Strokes in the quantum Otto cycle . 128 7.2.4 Harmonic oscillator working fluid with finite times . 130 7.2.5 Collective spin working fluid with finite times . 132 7.2.6 Analogous equilibrium cycle . 135 7.3 Quantum-mechanical evolution towards equilibrium . 136 8 Selected phenomena and applications 139 8.1 Reservoir models . 139 8.1.1 Tight-binding chain . 140 8.1.2 Example: Evolution under a 1d reservoir . 142 8.1.3 Higher-dimensional tight-binding lattices . 144 8.1.4 Reservoirs with periodic boundary conditions . 145 8.1.5 Spin reservoir: Ising model . 146 8.2 Topological single-particle pumping . 149 CONTENTS 5 Die Termine der Vorlesung sind Montags von 12:15{13:45 und Dienstags von 12:15{13:45. Die dazugeh¨orige Ubung¨ wird von Dr. Ling Na Wu gehalten. Wegen der Corona-Krise werden Vor- lesung und Ubung¨ voraussichtlich nur online stattfinden. Die Veranstaltung ist erweiterbar zu einem vollen Wahlpflichtfach (12 LP), indem zus¨atzlich an einem Seminar oder einer Spezialvor- lesung aus der Theoretischen Physik teilgenommen wird (in Absprache mit dem Dozenten). Voraussetzungen f¨urdie Teilnahme: Quantenmechanik, (w¨unschenswert: Quantenmechanik II), Spaß am T¨ufteln, Internetzugang Dieses Vorlesungsskript (Englisch) wird online verf¨ugbarsein unter http://www1.itp.tu-berlin.de/schaller/lectures.html. Korrekturen und Vorschl¨agesollten an folgende email-Adresse gesendet werden: [email protected]. Zuletzt noch ein Hinweis: Dieses Skript wird erst w¨ahrendder Vorlesung ausgebaut und verbessert werden. Es ist kein Originalwerk sondern basiert auf B¨uchern, Manuskripten, pers¨onlichen Erfahrungen und nicht zuletzt vielen wertvollen Hinweisen, ich danke hier z.B. insbesondere Philipp Stammer. Es wird nach jeder Vorlesung in aktualisierter Fassung online gestellt, es empfiehlt sich daher, nicht gleich zu Anfang alles auszudrucken. Literaturhinweise: M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cam- bridge University Press, Cambridge (2000). [1] J. Preskill, Quantum Computation, lecture notes http://www.theory.caltech.edu/~preskill/ ph219/. Wir werden folgende Inhalte behandeln klassische und Quanten-gates Quanten-Algorithmen Realisierungen adiabatische Quantenrechner Dekoh¨arenz 6 CONTENTS Prerequisites This lecture builds upon basic knowledge of quantum mechanics. You should therefore be familiar with the following: Quantum systems are described by the Schr¨odingerequation i~@tΨ(r; t) = H(r; t)Ψ(r; t) ; (1) where H(r; t) is known as Hamilton operator and may contain spatial derivatives acting to the right. The solution to this partial differential equation Ψ(r; t) is called wave function and has a probability density interpretation P (r; t) = jΨ(r; t)j2. We will make use of the Dirac Bra-Ket notation, where, notationally, the state of a quantum system can be characterized by a "ket" jΨ(t)i, which is a vector in a Hilbert space. Scalar products between two kets are denoted by hΦjΨi, and the wave function is then given by Ψ(r; t) = hrjΨ(t)i, with r^ jri = r jri denoting the eigenstates of the position operator. Throughout the lecture, we will use units where ~ = 1. In these units, energy has units of inverse time, such that H · t is manifestly dimensionless. The Schr¨odingerequation then reads for the state jΨ(t)i simply E _ Ψ = −iH jΨi : (2) Here, H = Hy is the Hamilton operator which can have different eigenstates. When H is time-independent, this equation is formally solved by −iHt jΨ(t)i = U(t) jΨ0i = e jΨ0i ; (3) where U(t) is known as time evolution operator and jΨ0i is the initial state. It is easy to show that it is unitary U y = U −1. When H is time-dependent, we can likewise introduce a time evolution operator that obeys jΨ(t)i = U(t) jΨ0i : (4) Insertion into the Schr¨odingerequation shows that the time evolution operator in general obeys d U(t) = −iH(t)U(t) : (5) dt 7 8 CONTENTS Integrating this equation yields the expansion Z t Z t Z t1 U(t) = 1 − i H(t1)dt1 − dt1 dt2H(t1)H(t2) ± ::: 0 0 0 1 Z t Z t1 Z tn−1 X n = (−i) dt1 dt2 ::: dtnH(t1) :::H(tn) n=0 0 0 0 1 n X Z t Z t ≡ T −i H(t0)dt0 = T exp −i H(t0)dt0 ; (6) n=0 0 0 where T is also known as time-ordering operator. By comparing the two expressions, one can see that T H(t1)H(t2) = H(t1)H(t2)Θ(t1 − t2) + H(t2)H(t1)Θ(t2 − t1), i.e., it sorts the operators by their time argument. The Hamiltonian of any two-level system (qubit) can be represented by Pauli matrices x y z H = hxσ + hyσ + hzσ ; 0 1 0 −i +1 0 σx = ; σy = ; σz = : (7) 1 0 +i 0 0 −1 y Since any Hamiltonian is hermitian H = H , we have that hα 2 R. The Pauli matrices obey the relations α β γ α β σ ; σ = 2iαβγσ ; σ ; σ = 2δαβ1 : (8) The time evolution operator for any two-level system subject to a constant Hamiltonian may be explicitly computed h hx x hy y hz zi −iHt −iht h σ + h σ + h σ U(t) = e = e = exp {−ihteh · σg = cos(ht)1 − i sin(ht)eh · σ ; (9) p 2 2 2 where we used that h = hx + hy + hz. Thereby, if we let a constant Hamiltonian H act for a certain time t, we can implement an arbitrary unitary operation on a two-level system. Upon a projective measurement of an observable O = Oy with spectral decomposition X O = λ` j`i h`j ; (10) ` with eigenvalues λ` and eigenstates j`i, we can only obtain measurement results λ`, i.e., the eigenvalues of the observable. Upon the outcome `, the state of the system instantaneously collapses to (`) j`i h`jΨi jΨi ! Ψ = ; (11) jh`jΨij 2 and the probability of this particular outcome is given by P` = jh`jΨij . This means that the measurement collapses the state of the system onto eigenstates of the observable. If we do not allow the state to change after the measurement (which happens e.g. if we measure the energy of the system and thereby collapse into an energy eigenstate), a subsequent measurement of the same observable will thus always yield the same result. Chapter 1 A brief intro to classical computation Why should we study classical computation if we are interested in quantum computation? For the circuit model, the answer is three-fold Many techniques and concepts from classical computation can be transferred to quantum computation. Computer scientists have thought about the resources it takes to solve a particular problem. They have invented classification schemes of difficult and not-so-difficult problems, and these schemes are useful to classify quantum algorithms as well. From knowing classical computation, we know where quantum computers may outperform classical ones.