Lecture 6: QUANTUM CIRCUITS
|
|
- Horatio Bradley
- 5 years ago
- Views:
Transcription
1 1. Simple Quantum Circuits Lecture 6: QUANTUM CIRCUITS We ve already mentioned the term quantum circuit. Now it is the time to provide a detailed look at quantum circuits because the term quantum computer itself is synonymous with the quantum circuit model of computation. Generally, a quantum circuit is formed by the gates connected by lines. The simplest quantum circuits containing the single qubit gates are shown in Figure 4.. Each line in the circuit represents a wire in the quantum circuit. This wire does not necessarily correspond to a physical wire. Instead, it can correspond simply to the passage of time, or perhaps to a particle such as a photon - a particle of light - moving from one location to another through space. The circuit is to be read from left-to-right. It is conventional to assume that the state input to the circuit is a computational basis state, usually the state consisting of all 0 >s. If this rule is broken, it is necessary to inform the user about the input state. The multi-qubit gates themselves may be represented by some circuits. For example, Figure 4.3 shows the circuit representation for the CONTROLLED-NOT gate. The top line represents the control qubit, the bottom line the target qubit. The symbol reminds about an addition modulo [see: Eq. (5.76)]. It is interesting what will happen if we change this symbol by the symbol for an arbitrary single qubit unitary operation U? Then we obtain the circuit drawn in Figure 4.4. The operation represented by such a circuit is said to be CONTROLLED-U operation. We see that this operation is again a two qubit operation, with a control and a targed qubit. If the control qubit is set to 1 > then U is applied to the target qubit, otherwise the targed qubit is left alone, that is a, b > a > U a b >. (1) Hence the CONTROLLED-U gate is a natural extension of the CONTROLLED- NOT gate, and the latter itself can be represented in a different way, as illustrated in Figure 1.9. We just consider the CONTROLLED-U operation with one target qubits. What about n target qubits? In circuit representation, the answer is obtained immediately: Figure 4.4 may be changed to Figure 1.8! The circuit representation can be used in order to implement an arbitrary CONTROLLED- U operation using only single qubit operations and the CONTROLLED-NOT gate. The example of such representation is on Figure 4.6, but we will not discuss it in detail. Using quantum circuits we can construct many other important operations. For example, the circuit in Figure 1.7 accomplishes a simple but very useful task - it swaps the states of the two qubits. To see this, let us write the sequence of effects of the gates shown in this Figure on a computational basis state a, b >: a, b > a, a b > a (a b), a b >= b, (a b) > b, (a b) b >= b, a >. () Here all additions are done modulo. The total effect of the circuit, therefore, is to interchange the states of the two qubits. An equivalent schematic symbol for this important operation is shown in Figure 1.7 on the right. 1
2 A slightly more complicated circuit, shown in Figure 1.1, has a Hadamard gate followed by a CONTROLLED NOT and transforms the four computational basis states according to the table given. We see that the final states are nothing but the Bell states which example we have discussed in one of our previous lectures [see: Eq. (4.17)]. Note the useful mnemonic rule for remembering all the Bell states (see test 4, question 4): β xy > 0y > +( 1)x 1, ȳ >, (3) where ȳ is the negation of y. So far we did not mention about the circuit representation of the measurement. Meanwhile, we know that the measurement plays the role of an interface between the quantum and classical worlds, and so it may present in any circuit designated for the solution of a practical task. To represent a usual projective measurement, a special meter symbol is introduced, as shown in Figure As previously described, this operation converts a single qubit state ψ >= c 0 0 > +c 1 1 > into a probabilistic classical bit M which is 0 with probability c 0 and 1 with probability c 1. To distinguish bit from qubit, the bit is drawing as a double-line wire (see Figure 1.10). Now we are well-experienced for understanding that quantum circuits can serve as models for all quantum processes which are in no way limited to only the specific problem of quantum computation. We shall illustrate this intriguing guess by considering two important examples in what follows.. No-Cloning Theorem As a first example, let us try to use the quantum CONTROLLED-NOT for copying a qubit in a way analogous that classical CONTROLLED-NOT does. This way is illustrated on the left-hand side of Figure A classical CONTROLLED- NOT is taken with the bit to copy (in some unknown state x) as a control bit and a scratchpad bit initialized to zero as a target bit. The output is two bits, both of which are in the same state x. In quantum case we have to copy a qubit in the unknown state ψ >= a 0 > +b 1 > (4) and to initiate a scratchpad qubit in the state 0 >. So the input state of these two qubits may be written as [ a 0 > +b 1 > ] 0 >= a 00 > +b 10 >. (5) The effect of the CONTROLLED-NOT is to negate the second qubit when the first qubit is 1, and hence the output state is ψ out = a 00 > +b 11 >. (6) Does it mean that the copying ψ > was successful? More specifically, have we created the state ψ > ψ > or not? We see that in particular cases where ψ >= 0 > or ψ >= 1 > (7)
3 that is indeed what this circuit does. Therefore, it is possible to use the quantum circuit shown in Figure 1.11, to copy information encoded as a 0 > or 1 >. But with the general state ψ > we have another story, since the output can be generally rewritten as ψ > ψ >= a 00 > +ab 01 > +ba 10 > +b 11 >. (8) Comparing (8) and (6), we see that unless ab = 0 the circuit above does not copy the quantum state input! We obtained a remarkable result: it turns out to be impossible to make a copy of an unknown quantum state. This result is known as the no-cloning theorem. 3. Quantum Teleportation Next we will apply the quantum circuit technique to clarify something very surprising and a lot of fun - quantum teleportation! Commonly, teleportation is understood as a fictional method for transferring an object between two places by a process of dissociation, information transmission and reconstitution. The net effect is the destruction of the original object at the source and the creation of an exact replica at the intended destination. A key feature of teleportation is that the actual object does not transverse the distance between two locations. Instead, the object is scanned to extract sufficient information to recreate the original, the information is transmitted and an exact replica is re-assembled at the destination out of the material is locally available. Of course, we are hardly at the point of being able to teleport in this manner an entire person, even though it will be possible in principle. Starting from the observation that, according to the data by the American National Institute of Health, it requires about 10 Gigabytes to hold the information needed to describe the three-dimensional structure of a human being to 1mm 3 resolution, it can be easily estimated that an entire human being could be described, down to the atomic level, using roughly 10 3 bits. With current communication channel capacities, it would like take about a hundred million centuries to transmit this information down a single channel. So, I leave funny stories for beaming action heroes around the Universe to video, and restrict myself only to the consideration of a small-scale teleportation prototype capable of teleporting a single qubit. Naturally, the prospects for a teleportation of such small objects as a qubit is much better from the information point of view. Nevertheless, until recently no serious attention has been paid to the physical principles on which such teleportation might be based. The arguments of most scientists were that the teleportation is impossible because, as the Heisenberg Uncertainty Principle stands, it is impossible to measure all the attributes of a quantum state exactly. For example, it is impossible to measure the position and momentum of a particle simultaneously. Consequently, it appeared that even a scanning step of teleportation is doomed to failure because it would never yield complete information about the original. The situation changed in 1993 when a team of of prominent physicist and computer scientists, Charles Bennet, Gilles Brassard and others, showed that it is possible to exploit yet another aspect of modern quantum theory, namely, the notion of entangled states and nonlocal interaction, to circumvent the limitations of the Heisenberg Uncertainty Principle and hence to create an exact replica of a quantum state. 3
4 In the context of a quantum teleportation, the entangled pair of particles serve as two ends of quantum communication channel: one particle being retained by the person wishing to teleport the quantum state and the other by the person wishing to receive it. Thus, in order to teleport a quantum state, the sender and recepient must each already possess one member of a pair of entangled particles. The basic idea is for the sender to make a measurement of the joint state of the particle whose state is to be teleported and one of the entangled particle, and then to send the result of the measurement as a classical message over a conventional communication channel (such as radio). For the recepient, it is to use the information in this classical message to determine which operation to apply to his member of the pair of entangled particles in order to place it in a state that is an exact replica of the state that the sender wished to teleport. Note that in this scheme the quantum state of an object is teleported, obviously not the object itself. Consequently, we cannot use this scheme to teleport an electron in its entirety from one place to another, but we can teleport, say, the spin orientation of one electron at a particular location to another electron at a different location. However, the net effect is similar: a particle in a specific state at the source place has its state destroyed and restored on another particle at the destination without the original particle traversing the intermediate distance. Now, we are able to give a more precise description of quantum teleportation represented by the quantum circuit shown in Figure The two top lines in this Figure represent sender s system, while the bottom line is the recepient s system. The state of the qubit to be sent is Ψ >= a 0 > +b 1 >. (9) The objective is to transmit this quantum state using classical bits and then to reconstruct the exact quantum state at the receiver. Of course, such a statement of the problem is surprising in light of the no-cloning principle because we want to transmit an unknown quantum state. Nevertheless, the problem has a solution. The key for teleportation is the use of the entangled particles in such a way that the first and second particle of the source EPR pair is accesible for the sender and receiver, respectively. Mathematically, the initial state of the EPR pair can be written as the joint two-particle state that cannot be factorized as the direct product of the states of two separate particles, say, β 00 >= 1 ( 00 > + 11 > ). (10) Until a particle is transmitted, only sender can perform transformation with the first particle and only receiver can perform transformation with the second particle in this joint state. At this point the qubit in the state (9) is not correlated yet with the entangled particles in the state (10), either classically or quantum mechanically, so we can still write the combined state for the initial three-particle system as a direct product of the states (9) and (10): Ψ 0 >= ψ > β 00 >= ( a 0 > +b 1 > ) ( ) 1 00 > + 11 > ( = 1 a 0 > ( 00 > + 11 > ) + b 1 > ( 00 > + 11 > ) = 1 ( a 000 > +a 011 > +b 100 > +b 111 > ). (11) 4
5 We recall that at this stage the sender controls the first two bits and the receiver the last bit in the three-qubit state (11). So, the sender is free to use first the CONTROLLED-NOT, obtaining the state ψ 1 = 1 ( a 000 > +a 011 > +b 110 > +b 101 > ) and then the Hadamard gate, obtaining the state ( ψ = 1 00 > ( a 0 > +b 1 > ) + 01 > ( a 1 > +b 0 > ) (1) + 10 > ( a 0 > b 1 > ) + 11 > ( a 1 > b 0 > )). (13) After this transformation, the sender measures the first two qubits to get one of the two-qubit states 00 >, 01 >, 10 > or 11 > with equal probability. Depending on the result of the measurement, the quantum state of receiver s qubit is projected to a 0 > +b 1 >, a 1 > +b 0 >, a 0 > b 1 > or a 1 > b 0 >, (14) respectively. That is, depending on sender s measurement outcome, receiver s qubit will end up in one of the following possible states 00 (a 0 > +b 1 >) ψ 3 (00) >; 01 (a 1 > +b 0 >) ψ 3 (01) >; 10 (a 0 > b 1 >) ψ 3 (10) >; 11 (a 1 > b 0 >) ψ 3 (11) >. (15) Of course, to know which state it is in, receiver must be told the result of sender s measurement using an ordinary classical information channel (a telephone, say). Once receiver has learned the measurement outcome, the state ψ > can be recovered by applying the appropriare quantum gate to the state of a qubit belonging to receiver. For example, in the case where the measurement yields 00, nothing needs to be done. If the measurement is 01, then receiver can fix up his state by applying the Z-gate. If the measurement is 11 then receiver can fix up his state by applying first an X and then a Z- gate. Summing up, receiver needs to apply the transformation Z M 1 X M to recover the state which has been send. It seems that a teleportation creates a copy of the quantum state being teleported - in apparent violation of the no-cloning theorem discussed in preceding section. However, this violation is only illusory since the original state Ψ >, which should to be sent to the receiver, is irretrievably altered under the measurement. That is, only target qubit is left in the state ψ >, while the second qubit ends up in one of the computational basis state 0 > or 1 >, depending upon the measurement result on the first qubit. Of course, later it recoveres due to the special procedures but be recovered is not the same as be a copy, isn t it? Another important aspect of a quantum teleportation must be commented: doesn t teleportation allow one to transmit quantum states faster than light? Fortunately, this is not the case because to complete the teleportation sender must transmit the measurement result to receiver over a classical communication channel which work, of course, is limited by the speed of light as it should be along the prescription of the theory of relativity. 5
6 It is worth noting, that if somebody can teleport a single qubit, an arbitrary message can be teleported by decomposing it into sequence of qubits and teleporting each qubit separately. So although the ability to teleport a single qubit may seem too modest, it actually provides the foundation for a reach new communication technology. We notice as well that quntum teleportation is closely related to quntum computing, since it might provide an alternative way for transmission of the primary quantum information inside a quantum computer or even between distant quantum computers. This might be especially useful if some qubits needs to be kept secret (as in quantum cryptography, for instance). Using quantum teleportation, a qubit could be passed around without ever being transmitted over insecure (public) channel. 4. Quantum dynamics: Schrödinger Equation Although the operator level of description is adequite for representing the transformation of states in many-qubit system, it tells us nothing about the time evolution that transforms the input state into the output one. To be able to consider this process, we need to understand a more general question of how a state vector evolves in time. As the state of quantum memory register is described by some state vector, this amounts to asking by what rule does the memory register of a quantum computer evolve? Fortunately, famous physicist Erwin Schrödinger already gave us an answer to this question back in 196, long before quantum computers were ever imagined. To understand this answer, we may return to the Dirac formulation of quantum mechanics adding the time parameter to the description of quantum states: Ψ > Ψ(t) >. (16) Then the fundamental postulate of quantum mechanics will sound as: The maximum information about the outcome of physical measurements at time t is contained in the probability amplitudes < l, m,... Ψ(t) >, which correspond to a complete set of observables L, M,... for the system. As expected, the only new feature here is that we now recognize formally that the state is a function of time. Furthermore, if we take two states, Ψ(t) > and Ψ(t 0 ) >, with t 0 < t, we shall assume that Ψ(t) > is determined by Ψ(t 0 ) > by means of some time-dependent operator U(t, t 0 ), Ψ(t) >= U(t, t 0 ) Ψ(t 0 ) >, (17) which legitimizes the name time development or evolution operator for U(t, t 0 ). This assertion expresses the general principle of causality in the quantum-mechanical form. If Eq. (17) takes place, we can immediately construct the time-dependent composition rule for the probability amplitudes: Ψ l (t) < l Ψ(t) >= m where the coefficients < l U(t, t 0 ) m >< m Ψ(t 0 ) > m S lm (t, t 0 ) < m Ψ(t 0 ) >, (18) S lm (t, t 0 ) =< l U(t, t 0 ) m > (19) are independent of the state Ψ(t 0 ) >. The physical interpretation of these coefficients is straightforward: they signifies the probability amplitude for finding the 6
7 system at time t in the eigenstate l > of the observable L, if at time t 0 the system was known to be in the eigenstate m > of the observable M. Hereafter, we will call these quantities the transition amplitudes. From (17) and (19) it follows that the time evolution operator U(t, t 0 ) has the following properties: U(t, t) = I; U(t, t 0 ) = U(t, t 1 )U(t 1, t 0 ); [U(t, t 0 )] 1 U 1 (t, t 0 ) = U(t 0, t). (0) Furthermore, for small ɛ we may write to first order U(t + ɛ, t 0 ) = U(t, t 0 ) + ɛ du(t, t 0) dt In particular case t 0 = t this takes the form U(t, t 0 ) ī h ɛh(t)u(t, t 0), (1) U(t + ɛ, t) = U(t, t) ī h ɛh(t)u(t, t) = [ I ī h ɛh(t)] U(t, t). () In general, Eq. () is equivalent to the differential equation for operator U(t, t 0 ), i h du(t, t 0) dt = H(t)U(t, t 0 ), (3) which should be solved with the initial condition U(t 0, t 0 ) = I. By this, we introduce new operator H(t) which is the same fundamental characteristic of the system as an original evolution operator U(t, t 0 ). The factor i/ h in this expression could look simply as a curious caprice if it were not be known the consequences of such presentation. To deduce these consequences, we consider the state or, to first order in ɛ, Ψ(t + ɛ) >= U(t + ɛ, t) Ψ(t) >, (4) Ψ(t) > +ɛ d dt Ψ(t) >= [ Hence, we derive the differential equation for the state vector, I ī h ɛh(t) ] Ψ(t) >. (5) i h d Ψ(t) >= H(t) Ψ(t) >. (6) dt which is the quantum-mechanical equation of motion in its most general form. We will call Eq. (6) the Schrödinger equation though primarily this equation has been suggested by Erwin Schrödinger not for the state vector but for the probability amplitude (or wave function ) in coordinate representation, i.e. for the function ψ(r, t) < r Ψ(t) >. (7) The most remarkable feature of the Schrödinger equation is the presence of a specific operator H(t) which is called the Hamiltonian. This operator relates to the total energy of the system which can be measured experimentally, and thus it 7
8 can be represented by Hermitian matrix. The form of this matrix is determined by the specific arrangement of atoms, molecules and charges that constitute the system (in particular, the computer). In the case of computer, we can think of the Hamiltonian as being analogous to the hardware and the initial state of the quantum memory register as being analogous to the data fed into a conventional computer. Of course, if quantum computer is specialized for performing only a single type of computation, the program is essentially folded into the definition of the Hamiltonian. We notice that due to the Hermitian property of the Hamiltonian H(t), the equations adjoint to Eqs. (3) and (6) becomes and i h du + (t, t 0 ) dt i h d dt = U + (t, t 0 )H(t) (8) < Ψ(t) =< Ψ(t) H(t) (9) By multiplying (3) on the left by U + (t, t 0 ) and (8) on the right by U(t, t 0 ), and subtracting the two resulting equations, we get d dt [U + (t, t 0 )U(t, t 0 )] = 0. (30) Therefore, the product of the operators conserves in time its value at t = t 0 equal to I, i.e. U + (t, t 0 )U(t, t 0 ) = I (31) for all t. Hence, the unitary operator is always unitary, and the norm of any state vector remains unchanged during the motion. Say, if < Ψ(t 0 ) Ψ(t 0 ) >= 1, then < Ψ(t) Ψ(t) >= 1 for all times t. Note that such superficially innocuous property of the evolution operator as its inherent unitarity harbors an extremely important implication. Namely, it means that the evolution operator of an ideal quantum computer, isolated from its enviroment, is reversible because the conjugate transpose of U(t, t 0 ) is equal to the inverse of it. Of course, quantum physicists knew all along that Schrödinger equation gives rise to a unitary (and hence reversible) evolution. Thus if a quantum system had any chance of serving as a computer, it had to be possible to make computers that operated reversibly. The importance of reversible computing has been first shown by Charles Bennett in Using the equations of motion (6) and (9), we are able to calculate the time derivative of the expectation value of an arbitrary operator A, which may itself vary with time: i h d A A dt < A >=< AH HA > +i h < [A, H] > +i h t t where < A >=< Ψ(t) A Ψ(t) >. where, as usual, the brackets signify expectation values of the operators enclosed. We see that if A commutes with H and is independent of time, the expectation value of A is constant, and A is said to be constant of motion. (3) 8
9 Finally, if the Hamiltonian of a computer is time independent, and its memory register is in the state Ψ(0) >, then the state of the memory register at an arbitrary time t is [ īh ] Ht Ψ(t) >= exp Ψ(0) >. (33) This state determines the general solution of the Schrödinger equation with the time-independent Hamiltonian and the initial state Ψ(0) >. APPENDIX: Simulation of Quantum Systems To close out our brief consideration of some interesting and useful applications of the quantum circuit model let cast for the simulation of the quantum systems. Of course, some quantum systems can be simulated by a classical computer. But their real quantum behavior might often be much more complex than its classical simulation. So any alternative ideas for the simulation of quantum systems are very welcome. The use of quantum computers for this purpose is one of the great challenges in modern science and technology. The heart of simulation of any system (classical or quantum) is the solution of differential equations which express the physical laws governing the dynamical behavior of a system. For quantum systems this is the Schrödinger equation (6). In many cases this equation takes the form of a partial differential equation like the well-known equation for a real particle moving in space in the presence of potential V (x), i h t ψ(x) = [ h m x + V (x)] ψ(x), (34) where ψ(x) is the wave function in the coordinate representation, ψ(x) =< x ψ >. (35) The latter is an elliptical equation very much like the wave equation. Therefore simulating the Schrödinger equation itself is not a special difficulty faced in simulating quantum systems since its solution can be obtain, as usual, by approximating the state with a digital representation, then discretizing the differential equation in space and time such that an iterative application of a procedure carries the state from the initial to the final conditions. By this sequence of steps we achieve the goal which is generally formulated as: what is the state at some other time and/or position, given an initial state of the system? What is then the difficulty? The key challenge in simulating quantum systems is the exponential number of differential equations which must be solved. Say, for one qubit evolving according to the the Schrödinger equation, a system of two differential equations must be solved; for two qubits, four equations; and for n qubits, n equations. Sometimes, insightful approximations can be made which reduce the effective number of equations involved, thus making classical simulation of the quantum system feasible. For example, in many cases the Hamiltonian can be represented as a sum over many local interactions. Specifically, for a system of n particles L H = H k, (36) k=1 where each H k acts on at most a constant c number of systems, and L is a polynomial in n. Moreover, the terms H k are often just two-body interactions. Such locality 9
10 is quite physically reasonable, and originates in many systems from the fact that the interactions mostly fall off with increasing distance or difference in energy. The important point is that usually e iht is difficult to compute, while e ih kt acts on a much smaller subsystem, and is straightforward to approximate using quantum circuits. However, it is strongly desired to have the quantum circuit which allows one an efficient quantum simulating even Hamiltonians which are, by definition, not the sum of local interactions. An example of such quantum circuit is shown in Figure This circuit performs simulating the Hamiltonian H = Z 1 Z Z 3, (37) which acts on a three qubit system. This circuit implements e ih t for arbitrary values of t. The main observation is that although the Hamiltonian involves all the qubits in the system, it does so in a classical manner: the phase shift applied to the system is e i t if the parity of the qubits in the computational basis is even; otherwise, the phase shift should be e i t. Thus simple simulation of H is achieved by first classically computing the parity (storing the result in an ancillary qubit), then applying the appropriate phase shift conditioned on the parity, and then uncomputing the parity (to erase the ancillary qubit). This strategy clearly works not only for three qubits, but also for arbitrary number of qubits n. As a result, we can use this procedure for efficient simulating more complicated Hamiltonians of the form H = σ 1 c(1) σ c()... σ n c(n), (38) where σc(k) k is a Pauli matrix acting on the kth qubit, with c(k) {0, 1,, 3} specifying one of the operations {I, X, Y, Z)}. Indeed, the qubits upon which the identity operation is performed can be disregarded, and X or Y terms can be transformed, as we early saw, to Z operations by single qubit gates. This leaves us with a Hamiltonian of the form H = Z 1 Z... Z n, (39) which is extension of the Hamiltonian (37) for arbitrary number of operators n. 10
Lecture 4. QUANTUM MECHANICS FOR MULTIPLE QUBIT SYSTEMS
Lecture 4. QUANTUM MECHANICS FOR MULTIPLE QUBIT SYSTEMS 4.1 Multiple Qubits Next we consider a system of two qubits. If these were two classical bits, then there would be four possible states,, 1, 1, and
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
More informationHilbert Space, Entanglement, Quantum Gates, Bell States, Superdense Coding.
CS 94- Bell States Bell Inequalities 9//04 Fall 004 Lecture Hilbert Space Entanglement Quantum Gates Bell States Superdense Coding 1 One qubit: Recall that the state of a single qubit can be written as
More informationA review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels
JOURNAL OF CHEMISTRY 57 VOLUME NUMBER DECEMBER 8 005 A review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels Miri Shlomi
More informationQuantum Gates, Circuits & Teleportation
Chapter 3 Quantum Gates, Circuits & Teleportation Unitary Operators The third postulate of quantum physics states that the evolution of a quantum system is necessarily unitary. Geometrically, a unitary
More informationb) (5 points) Give a simple quantum circuit that transforms the state
C/CS/Phy191 Midterm Quiz Solutions October 0, 009 1 (5 points) Short answer questions: a) (5 points) Let f be a function from n bits to 1 bit You have a quantum circuit U f for computing f If you wish
More informationUnitary Dynamics and Quantum Circuits
qitd323 Unitary Dynamics and Quantum Circuits Robert B. Griffiths Version of 20 January 2014 Contents 1 Unitary Dynamics 1 1.1 Time development operator T.................................... 1 1.2 Particular
More informationQuantum Computing: Foundations to Frontier Fall Lecture 3
Quantum Computing: Foundations to Frontier Fall 018 Lecturer: Henry Yuen Lecture 3 Scribes: Seyed Sajjad Nezhadi, Angad Kalra Nora Hahn, David Wandler 1 Overview In Lecture 3, we started off talking about
More informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 03/19/2018 Postulates of Quantum Mechanics The postulates (axioms) of quantum mechanics
More informationE = hν light = hc λ = ( J s)( m/s) m = ev J = ev
Problem The ionization potential tells us how much energy we need to use to remove an electron, so we know that any energy left afterwards will be the kinetic energy of the ejected electron. So first we
More informationIntroduction to Quantum Computing for Folks
Introduction to Quantum Computing for Folks Joint Advanced Student School 2009 Ing. Javier Enciso encisomo@in.tum.de Technische Universität München April 2, 2009 Table of Contents 1 Introduction 2 Quantum
More informationQuantum Error Correcting Codes and Quantum Cryptography. Peter Shor M.I.T. Cambridge, MA 02139
Quantum Error Correcting Codes and Quantum Cryptography Peter Shor M.I.T. Cambridge, MA 02139 1 We start out with two processes which are fundamentally quantum: superdense coding and teleportation. Superdense
More informationLecture 2: Introduction to Quantum Mechanics
CMSC 49: Introduction to Quantum Computation Fall 5, Virginia Commonwealth University Sevag Gharibian Lecture : Introduction to Quantum Mechanics...the paradox is only a conflict between reality and your
More informationLecture 11 September 30, 2015
PHYS 7895: Quantum Information Theory Fall 015 Lecture 11 September 30, 015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
More informationAn Introduction to Quantum Information. By Aditya Jain. Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata
An Introduction to Quantum Information By Aditya Jain Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata 1. Introduction Quantum information is physical information that is held in the state of
More informationSUPERDENSE CODING AND QUANTUM TELEPORTATION
SUPERDENSE CODING AND QUANTUM TELEPORTATION YAQIAO LI This note tries to rephrase mathematically superdense coding and quantum teleportation explained in [] Section.3 and.3.7, respectively (as if I understood
More information226 My God, He Plays Dice! Entanglement. Chapter This chapter on the web informationphilosopher.com/problems/entanglement
226 My God, He Plays Dice! Entanglement Chapter 29 20 This chapter on the web informationphilosopher.com/problems/entanglement Entanglement 227 Entanglement Entanglement is a mysterious quantum phenomenon
More informationShort introduction to Quantum Computing
November 7, 2017 Short introduction to Quantum Computing Joris Kattemölle QuSoft, CWI, Science Park 123, Amsterdam, The Netherlands Institute for Theoretical Physics, University of Amsterdam, Science Park
More informationLecture 6: Quantum error correction and quantum capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde The quantum capacity theorem is one of the most important theorems in quantum hannon theory. It is a fundamentally quantum theorem
More informationThe Hamiltonian and the Schrödinger equation Consider time evolution from t to t + ɛ. As before, we expand in powers of ɛ; we have. H(t) + O(ɛ 2 ).
Lecture 12 Relevant sections in text: 2.1 The Hamiltonian and the Schrödinger equation Consider time evolution from t to t + ɛ. As before, we expand in powers of ɛ; we have U(t + ɛ, t) = I + ɛ ( īh ) H(t)
More informationErrata list, Nielsen & Chuang. rrata/errata.html
Errata list, Nielsen & Chuang http://www.michaelnielsen.org/qcqi/errata/e rrata/errata.html Part II, Nielsen & Chuang Quantum circuits (Ch 4) SK Quantum algorithms (Ch 5 & 6) Göran Johansson Physical realisation
More informationQuantum Information Types
qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information, Phys. Rev. A 76 (2007) 062320; arxiv:0707.3752 Contents 1 Introduction
More informationSeminar 1. Introduction to Quantum Computing
Seminar 1 Introduction to Quantum Computing Before going in I am also a beginner in this field If you are interested, you can search more using: Quantum Computing since Democritus (Scott Aaronson) Quantum
More informationWeek 11: April 9, The Enigma of Measurement: Detecting the Quantum World
Week 11: April 9, 2018 Quantum Measurement The Enigma of Measurement: Detecting the Quantum World Two examples: (2) Measuring the state of electron in H-atom Electron can be in n = 1, 2, 3... state. In
More informationQuantum Computers. Todd A. Brun Communication Sciences Institute USC
Quantum Computers Todd A. Brun Communication Sciences Institute USC Quantum computers are in the news Quantum computers represent a new paradigm for computing devices: computers whose components are individual
More informationEnsembles and incomplete information
p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system
More information6.080/6.089 GITCS May 6-8, Lecture 22/23. α 0 + β 1. α 2 + β 2 = 1
6.080/6.089 GITCS May 6-8, 2008 Lecturer: Scott Aaronson Lecture 22/23 Scribe: Chris Granade 1 Quantum Mechanics 1.1 Quantum states of n qubits If you have an object that can be in two perfectly distinguishable
More informationChecking Consistency. Chapter Introduction Support of a Consistent Family
Chapter 11 Checking Consistency 11.1 Introduction The conditions which define a consistent family of histories were stated in Ch. 10. The sample space must consist of a collection of mutually orthogonal
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationEntanglement and information
Ph95a lecture notes for 0/29/0 Entanglement and information Lately we ve spent a lot of time examining properties of entangled states such as ab è 2 0 a b è Ý a 0 b è. We have learned that they exhibit
More information1 Measurements, Tensor Products, and Entanglement
Stanford University CS59Q: Quantum Computing Handout Luca Trevisan September 7, 0 Lecture In which we describe the quantum analogs of product distributions, independence, and conditional probability, and
More information1. Basic rules of quantum mechanics
1. Basic rules of quantum mechanics How to describe the states of an ideally controlled system? How to describe changes in an ideally controlled system? How to describe measurements on an ideally controlled
More informationCSCI 2570 Introduction to Nanocomputing. Discrete Quantum Computation
CSCI 2570 Introduction to Nanocomputing Discrete Quantum Computation John E Savage November 27, 2007 Lect 22 Quantum Computing c John E Savage What is Quantum Computation It is very different kind of computation
More informationLecture 3: Hilbert spaces, tensor products
CS903: Quantum computation and Information theory (Special Topics In TCS) Lecture 3: Hilbert spaces, tensor products This lecture will formalize many of the notions introduced informally in the second
More informationQuantum Cryptography
Quantum Cryptography (Notes for Course on Quantum Computation and Information Theory. Sec. 13) Robert B. Griffiths Version of 26 March 2003 References: Gisin = N. Gisin et al., Rev. Mod. Phys. 74, 145
More informationLecture 1: Introduction to Quantum Computing
Lecture 1: Introduction to Quantum Computing Rajat Mittal IIT Kanpur Whenever the word Quantum Computing is uttered in public, there are many reactions. The first one is of surprise, mostly pleasant, and
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationQuantum Teleportation
Fortschr. Phys. 50 (2002) 5 7, 608 613 Quantum Teleportation Samuel L. Braunstein Informatics, Bangor University, Bangor LL57 1UT, UK schmuel@sees.bangor.ac.uk Abstract Given a single copy of an unknown
More information. Here we are using the standard inner-product over C k to define orthogonality. Recall that the inner-product of two vectors φ = i α i.
CS 94- Hilbert Spaces, Tensor Products, Quantum Gates, Bell States 1//07 Spring 007 Lecture 01 Hilbert Spaces Consider a discrete quantum system that has k distinguishable states (eg k distinct energy
More informationPLEASE LET ME KNOW IF YOU FIND TYPOS (send to
Teoretisk Fysik KTH Advanced QM (SI2380), Lecture 2 (Summary of concepts) 1 PLEASE LET ME KNOW IF YOU FIND TYPOS (send email to langmann@kth.se) The laws of QM 1. I now discuss the laws of QM and their
More informationSecurity Implications of Quantum Technologies
Security Implications of Quantum Technologies Jim Alves-Foss Center for Secure and Dependable Software Department of Computer Science University of Idaho Moscow, ID 83844-1010 email: jimaf@cs.uidaho.edu
More informationVector Spaces in Quantum Mechanics
Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.
More informationHardy s Paradox. Chapter Introduction
Chapter 25 Hardy s Paradox 25.1 Introduction Hardy s paradox resembles the Bohm version of the Einstein-Podolsky-Rosen paradox, discussed in Chs. 23 and 24, in that it involves two correlated particles,
More informationQuantum Measurements: some technical background
Quantum Measurements: some technical background [From the projection postulate to density matrices & (introduction to) von Neumann measurements] (AKA: the boring lecture) First: One more example I wanted
More informationLecture 1: Introduction to Quantum Computing
Lecture : Introduction to Quantum Computing Rajat Mittal IIT Kanpur What is quantum computing? This course is about the theory of quantum computation, i.e., to do computation using quantum systems. These
More informationLecture 3. QUANTUM MECHANICS FOR SINGLE QUBIT SYSTEMS 1. Vectors and Operators in Quantum State Space
Lecture 3. QUANTUM MECHANICS FOR SINGLE QUBIT SYSTEMS 1. Vectors and Operators in Quantum State Space The principles of quantum mechanics and their application to the description of single and multiple
More informationTransmitting and Hiding Quantum Information
2018/12/20 @ 4th KIAS WORKSHOP on Quantum Information and Thermodynamics Transmitting and Hiding Quantum Information Seung-Woo Lee Quantum Universe Center Korea Institute for Advanced Study (KIAS) Contents
More informationQuantum information and quantum computing
Middle East Technical University, Department of Physics January 7, 009 Outline Measurement 1 Measurement 3 Single qubit gates Multiple qubit gates 4 Distinguishability 5 What s measurement? Quantum measurement
More informationSecrets of Quantum Information Science
Secrets of Quantum Information Science Todd A. Brun Communication Sciences Institute USC Quantum computers are in the news Quantum computers represent a new paradigm for computing devices: computers whose
More informationModern Physics notes Spring 2007 Paul Fendley Lecture 27
Modern Physics notes Spring 2007 Paul Fendley fendley@virginia.edu Lecture 27 Angular momentum and positronium decay The EPR paradox Feynman, 8.3,.4 Blanton, http://math.ucr.edu/home/baez/physics/quantum/bells
More informationUnitary evolution: this axiom governs how the state of the quantum system evolves in time.
CS 94- Introduction Axioms Bell Inequalities /7/7 Spring 7 Lecture Why Quantum Computation? Quantum computers are the only model of computation that escape the limitations on computation imposed by the
More informationLecture 2: From Classical to Quantum Model of Computation
CS 880: Quantum Information Processing 9/7/10 Lecture : From Classical to Quantum Model of Computation Instructor: Dieter van Melkebeek Scribe: Tyson Williams Last class we introduced two models for deterministic
More informationQuantum Entanglement and Cryptography. Deepthi Gopal, Caltech
+ Quantum Entanglement and Cryptography Deepthi Gopal, Caltech + Cryptography Concisely: to make information unreadable by anyone other than the intended recipient. The sender of a message scrambles/encrypts
More informationExample: sending one bit of information across noisy channel. Effects of the noise: flip the bit with probability p.
Lecture 20 Page 1 Lecture 20 Quantum error correction Classical error correction Modern computers: failure rate is below one error in 10 17 operations Data transmission and storage (file transfers, cell
More informationQubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable,
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable, A qubit: a sphere of values, which is spanned in projective sense by two quantum
More informationChapter 13: Photons for quantum information. Quantum only tasks. Teleportation. Superdense coding. Quantum key distribution
Chapter 13: Photons for quantum information Quantum only tasks Teleportation Superdense coding Quantum key distribution Quantum teleportation (Theory: Bennett et al. 1993; Experiments: many, by now) Teleportation
More informationStochastic Processes
qmc082.tex. Version of 30 September 2010. Lecture Notes on Quantum Mechanics No. 8 R. B. Griffiths References: Stochastic Processes CQT = R. B. Griffiths, Consistent Quantum Theory (Cambridge, 2002) DeGroot
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationSinglet State Correlations
Chapter 23 Singlet State Correlations 23.1 Introduction This and the following chapter can be thought of as a single unit devoted to discussing various issues raised by a famous paper published by Einstein,
More informationWhat is a quantum computer? Quantum Architecture. Quantum Mechanics. Quantum Superposition. Quantum Entanglement. What is a Quantum Computer (contd.
What is a quantum computer? Quantum Architecture by Murat Birben A quantum computer is a device designed to take advantage of distincly quantum phenomena in carrying out a computational task. A quantum
More informationCS257 Discrete Quantum Computation
CS57 Discrete Quantum Computation John E Savage April 30, 007 Lect 11 Quantum Computing c John E Savage Classical Computation State is a vector of reals; e.g. Booleans, positions, velocities, or momenta.
More informationBits. Chapter 1. Information can be learned through observation, experiment, or measurement.
Chapter 1 Bits Information is measured in bits, just as length is measured in meters and time is measured in seconds. Of course knowing the amount of information is not the same as knowing the information
More informationPhysics is becoming too difficult for physicists. David Hilbert (mathematician)
Physics is becoming too difficult for physicists. David Hilbert (mathematician) Simple Harmonic Oscillator Credit: R. Nave (HyperPhysics) Particle 2 X 2-Particle wave functions 2 Particles, each moving
More informationStochastic Histories. Chapter Introduction
Chapter 8 Stochastic Histories 8.1 Introduction Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in
More information2 Quantum Mechanics. 2.1 The Strange Lives of Electrons
2 Quantum Mechanics A philosopher once said, It is necessary for the very existence of science that the same conditions always produce the same results. Well, they don t! Richard Feynman Today, we re going
More informationPage 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02
Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation
More informationQuantum Computers. Peter Shor MIT
Quantum Computers Peter Shor MIT 1 What is the difference between a computer and a physics experiment? 2 One answer: A computer answers mathematical questions. A physics experiment answers physical questions.
More informationLecture 20: Bell inequalities and nonlocality
CPSC 59/69: Quantum Computation John Watrous, University of Calgary Lecture 0: Bell inequalities and nonlocality April 4, 006 So far in the course we have considered uses for quantum information in the
More informationMany Body Quantum Mechanics
Many Body Quantum Mechanics In this section, we set up the many body formalism for quantum systems. This is useful in any problem involving identical particles. For example, it automatically takes care
More informationImplementing Competitive Learning in a Quantum System
Implementing Competitive Learning in a Quantum System Dan Ventura fonix corporation dventura@fonix.com http://axon.cs.byu.edu/dan Abstract Ideas from quantum computation are applied to the field of neural
More informationQuantum Teleportation Pt. 3
Quantum Teleportation Pt. 3 PHYS 500 - Southern Illinois University March 7, 2017 PHYS 500 - Southern Illinois University Quantum Teleportation Pt. 3 March 7, 2017 1 / 9 A Bit of History on Teleportation
More informationCSE 599d - Quantum Computing Fault-Tolerant Quantum Computation and the Threshold Theorem
CSE 599d - Quantum Computing Fault-Tolerant Quantum Computation and the Threshold Theorem Dave Bacon Department of Computer Science & Engineering, University of Washington In the last few lectures, we
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 1: Quantum circuits and the abelian QFT
Quantum algorithms (CO 78, Winter 008) Prof. Andrew Childs, University of Waterloo LECTURE : Quantum circuits and the abelian QFT This is a course on quantum algorithms. It is intended for graduate students
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationEntanglement and Quantum Teleportation
Entanglement and Quantum Teleportation Stephen Bartlett Centre for Advanced Computing Algorithms and Cryptography Australian Centre of Excellence in Quantum Computer Technology Macquarie University, Sydney,
More information1.1 The Boolean Bit. Note some important things about information, some of which are illustrated in this example:
Chapter Bits Information is measured in bits, just as length is measured in meters and time is measured in seconds. Of course knowing the amount of information, in bits, is not the same as knowing the
More information15 Skepticism of quantum computing
15 Skepticism of quantum computing Last chapter, we talked about whether quantum states should be thought of as exponentially long vectors, and I brought up class BQP/qpoly and concepts like quantum advice.
More informationInstantaneous Nonlocal Measurements
Instantaneous Nonlocal Measurements Li Yu Department of Physics, Carnegie-Mellon University, Pittsburgh, PA July 22, 2010 References Entanglement consumption of instantaneous nonlocal quantum measurements.
More informationCSE 599d - Quantum Computing The No-Cloning Theorem, Classical Teleportation and Quantum Teleportation, Superdense Coding
CSE 599d - Quantum Computing The No-Cloning Theorem, Classical Teleportation and Quantum Teleportation, Superdense Coding Dave Bacon Department of Computer Science & Engineering, University of Washington
More informationThe quantum state as a vector
The quantum state as a vector February 6, 27 Wave mechanics In our review of the development of wave mechanics, we have established several basic properties of the quantum description of nature:. A particle
More informationQuantum Algorithms. Andreas Klappenecker Texas A&M University. Lecture notes of a course given in Spring Preliminary draft.
Quantum Algorithms Andreas Klappenecker Texas A&M University Lecture notes of a course given in Spring 003. Preliminary draft. c 003 by Andreas Klappenecker. All rights reserved. Preface Quantum computing
More informationTutorial on Quantum Computing. Vwani P. Roychowdhury. Lecture 1: Introduction
Tutorial on Quantum Computing Vwani P. Roychowdhury Lecture 1: Introduction 1 & ) &! # Fundamentals Qubits A single qubit is a two state system, such as a two level atom we denote two orthogonal states
More informationC/CS/Phys 191 Quantum Gates and Universality 9/22/05 Fall 2005 Lecture 8. a b b d. w. Therefore, U preserves norms and angles (up to sign).
C/CS/Phys 191 Quantum Gates and Universality 9//05 Fall 005 Lecture 8 1 Readings Benenti, Casati, and Strini: Classical circuits and computation Ch.1.,.6 Quantum Gates Ch. 3.-3.4 Universality Ch. 3.5-3.6
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More informationQuantum computation: a tutorial
Quantum computation: a tutorial Samuel L. Braunstein Abstract: Imagine a computer whose memory is exponentially larger than its apparent physical size; a computer that can manipulate an exponential set
More informationUniversal quantum computers
F. UNIVERSAL QUANTUM COMPUTERS 169 F Universal quantum computers Hitherto we have used a practical definition of universality: since conventional digital computers are implemented in terms of binary digital
More informationThe query register and working memory together form the accessible memory, denoted H A. Thus the state of the algorithm is described by a vector
1 Query model In the quantum query model we wish to compute some function f and we access the input through queries. The complexity of f is the number of queries needed to compute f on a worst-case input
More informationDelayed Choice Paradox
Chapter 20 Delayed Choice Paradox 20.1 Statement of the Paradox Consider the Mach-Zehnder interferometer shown in Fig. 20.1. The second beam splitter can either be at its regular position B in where the
More information1 1D Schrödinger equation: Particle in an infinite box
1 OF 5 1 1D Schrödinger equation: Particle in an infinite box Consider a particle of mass m confined to an infinite one-dimensional well of width L. The potential is given by V (x) = V 0 x L/2, V (x) =
More informationQuantum Entanglement. Chapter Introduction. 8.2 Entangled Two-Particle States
Chapter 8 Quantum Entanglement 8.1 Introduction In our final chapter on quantum mechanics we introduce the concept of entanglement. This is a feature of two-particle states (or multi-particle states) in
More information09a. Collapse. Recall: There are two ways a quantum state can change: 1. In absence of measurement, states change via Schrödinger dynamics:
09a. Collapse Recall: There are two ways a quantum state can change:. In absence of measurement, states change via Schrödinger dynamics: ψ(t )!!! ψ(t 2 ) Schrödinger evolution 2. In presence of measurement,
More informationDC metrology. Resources and methods for learning about these subjects (list a few here, in preparation for your research):
DC metrology This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationRichard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo
CS 497 Frontiers of Computer Science Introduction to Quantum Computing Lecture of http://www.cs.uwaterloo.ca/~cleve/cs497-f7 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum
More informationLecture 21: Quantum communication complexity
CPSC 519/619: Quantum Computation John Watrous, University of Calgary Lecture 21: Quantum communication complexity April 6, 2006 In this lecture we will discuss how quantum information can allow for a
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationCS/Ph120 Homework 1 Solutions
CS/Ph0 Homework Solutions October, 06 Problem : State discrimination Suppose you are given two distinct states of a single qubit, ψ and ψ. a) Argue that if there is a ϕ such that ψ = e iϕ ψ then no measurement
More informationA Simple Model of Quantum Trajectories. Todd A. Brun University of Southern California
A Simple Model of Quantum Trajectories Todd A. Brun University of Southern California Outline 1. Review projective and generalized measurements. 2. A simple model of indirect measurement. 3. Weak measurements--jump-like
More informationPhysical Systems. Chapter 11
Chapter 11 Physical Systems Until now we have ignored most aspects of physical systems by dealing only with abstract ideas such as information. Although we assumed that each bit stored or transmitted was
More information