Linear algebra for quantum theory pdf

Note that this is done by a single application of the circuit in Fig. Work out the quantum circuit depicted in Fig. Since p can be any number between The algorithm is decomposed into the following steps: 1. Now we measure the second register. In fact, we do not need to know the measurement outcome.

Of course, this equation is not enough to identify the period p. The number of trials necessary for this is O n with a good probability. References [1] Z. Deutsch, Proc. A, , 97 Deutsch and R. Jozsa, Proc.

A, , Bernstein and U. Simon, Proc. Press, Los Alamitos, Mihara and S. Sung, Comput. The second register is measured only for projection purposes, and reading the outcome is not nec- essary. Both of them depend on quantum integral transforms, which will be introduced in the present chapter. We mainly follow [1] in our presentation.

By substituting Eq. There are exponentially large numbers 2n of y for an n- qubit register, and this fact provides a quantum computer with exponentially fast computing power for a certain kind of computations compared to classical alternatives. The unitary matrix U implementing a discrete integral transform as in Eq.

The following example is taken from [2]. Let f x be a periodic function with the period P. Let us introduce an important gate, the controlled-Bjk gate. The inverted controlled-Bjk gate b and the controlled-Bjk gate are equivalent see Lemma 6. LEMMA 6. These results may be summarized as Eq. Note also that, in spite of its tensor product looking appearance, the last line of Eq.

Equation 6. Before writing down the quantum circuit realizing Eq. The reader should verify the above implementation by explicitly writing down the gates as matrices. Again note here that we should be careful in ordering the gates so that the control bit xj acts in Ujk before it is acted by a Hadamard gate. Since Eq. The equation that generalizes Eq. Proposition 4. The Walsh-Hadamard transform, written in the form of Eq. Quantum Integral Transforms 6.

The kernel K2 has been given above. Thus these gates are implemented with the set of universal gates. In this case K3 is written as a product of four two-level unitary matrices. References [1] Y. Berman, G. Doolen, R. Mainieri and V. It is required to take O N steps on average if a classical algorithm is employed.

Our presentation in this chapter closely follows [4] and [5]. In mathematical language, this is expressed as follows. It is assumed that f x is instantaneously calculable, such that this process does not require any computational steps.

A function of this sort is often called an oracle as noted in Chapter 5. We describe the algorithm in several steps. Let us evaluate the matrix elements of the RHS of Eq. Equation 7. The recursion relations 7. Figure 7. The number k of iteration is a 1, b 2, c 3 and d 4. We gave working space for oracles explicitly.

The box Uf is repeated m times to maximize Pz,k. The gate Rf is the oracle, and working qubits to implement the oracle are given explicitly. The subset A is of course unknown to us beforehand. References [1] L. Grover, Proc. Grover, Phys. Grover, Amer. Although the factorization algorithm may be carried out with a classical computer, it takes an exponentially longer time i.

Our presentation here closely follows [3] and [4]. The table below shows how long it takes for this PC to factor a large integer N by using FactorInteger commmand of Mathematica; N time[s] 0. The RSA cryptography [6] makes use of this fact to encode and decode messages. Bob wants to send Alice a message through a public communication channel. He encrypts his message with a key Alice publicizes. Although the key is publicly available, Alice is the only person who can decode the message.

Alice prepares two large prime numbers p and q, which she keeps secret and publishes the product N of them. It takes quite long in practical situations with more digits for N to factor N into p and q. This number e is also published along with N.

Alice keeps d secret. We outline how the RSA system works below. Equation 8. Let 8. Now suppose m is not a multiple of p. The RSA cryptosystem depends heavily on the belief that factorization of a large number into its prime factors is practically impossible. It turns out that the following algorithm is best suited for our purpose. Calculate the greatest common divisor gcd m, N by the Euclidean algorithm.

The number P is called the order or period. A quantum computer is required only in this step, and the rest may be executed in polynomial steps even with a classical computer.

If P is even, proceed to step 4. So this is OK. Of course we have cheated here and a quantum computer must be used for a large N. Let us proceed to step 4. There are m whose orders are less than Good luck!

STEP 2. Apply Uf on the state prepared in step 2. Find n which satisifes Eq. Find also the order P. Q2 P 2 Proof. Since m and N are coprime, so are mb and N. Q2 P 2 P Figure 8. Figure 8. Needless to say, this strategy is not practical when N is considerably large. The quantum circuit in Fig. However, the number of measurements required to guess P grows rapidly as N becomes larger and larger.

For example, "4. Let us consider continued fraction expansion of a rational number. Continued fraction expansion exists also for an irrational number, but it does not terminate. Let x be a rational number to be expanded. The M th convergent is x itself. Note that [a0 , a1 ,.

Thus the number M may be made either even or odd. Find the continued fraction expansion [a0 , a1 ,. We obtain the sequence p0 , q0 , p1 , q1 ,. Such k is unique. Find the order P by repeating the above algorithm. Apply the above algorithm. We must know when we obtain the correct order and when not. It is important to note that it is impossible to tell whether a particular outcome y is in C or not since P is not known in advance.

We prove this by induction. Suppose [a0 , a1 ,. Then [a0 , a1 ,. LEMMA 8. We assume, without loss of generality, that m is even. It should be also noted that there may be convergents of x, which do not satisfy the above inequality.

Suppose there are two sets of d, P satisfying this condition, which we call d1 , P1 and d2 , P2. Then the number P obtained by the algorithm in this section is the correct order of the modular exponential function mx mod N. Let [a0 , a1 ,. P Q 2Q Such d must be unique due to Lemma 8. However, Lemma 8. We have seen in Example 8. It should be kept in mind that P , and hence C, is not known in advance.

Here we follow the standard implementation given in [7] and [4]. Implementation of a modular exponential function is divided into several steps. We need to implement 1. Modular multiplexer, which outputs ab mod N. Modular exponential function, which outputs mx mod N. Let ck denote the carry bit. It is instructive to examine how two-digit numbers are added quantum mechanically.

Now we want to implement a quantum circuit, which we call ADD 2 , carrying out the above algebra. Our implementation is generalized to an n-qubit adder ADD n subsequently.

We have to align qubits in such a way that the gate acts on them in a nice way. It will be also denoted as a black box called SUM. We drop all the subscripts to simplify our notation hereafter. Let us make sure it works OK. We have explicitly written nontrivial gates only, and all the other qubits are acted by the unit matrix I as before.

Note that carry bits are required to compute si. We need b1 for the next step. Before we verify the circuit in Fig.

We denote the unitary matrix correspoinding to the kth layer by Uk. The adder for n-bit numbers, which we call ADD n , is obtained immedi- ately by generalizing the above implementation. We need to calculate all the carry bits ci for this purpose, and the left layers of the gates before sn is obtained are devoted for calculating the carry bits.

The readers should verify the circuit indeed implements n-bit addition. We need to introduce an n-qubit subtraction circuit to this end. Note the order of the input bits and the output bits. Uk denotes the unitary operation of the kth layer. Verify that s2 in Eq. Now we are ready to implement the modular adder. Let us verify its operations.

Therefore a and N are hardwired as parts of the circuit, while x is one of the input parameters. Let us verify it works as expected. The output of the second register is ax mod N. This circuit is denoted as b , where the temporary register has no external input and output ports and is not shown explicitly. We need to implement a quantum circuit which outputs ax mod N. Let us verify how it works to produce the correct result.

The numbers a and N are hardwired. The result of the modular exponential function is stored in the second register. The evaluation is divided into several steps. The power of polynomial depends on the actual implementation. Implementation with a mini- mal number of qubits requires O n2 elementary gates, but it may be reduced to O n if some extra qubits are added. The maximum power of the polynomial depends on actual circuit implementation and algorithms em- ployed to design the circuit.

It may happen that adding extra qubits makes the maximum power smaller. It was also shown in Chapter 6 that the quantum Fourier transform circuit is implemented with O n2 number of elementary gates. Shor, Proc. Lomonaco, Jr. Bornemann, Notices of the AMS, 50, , for example. Rivest, A. Shamir and L. Adleman, Comm.

ACM, 21, Vedral, A. Barenco and A. A 54, This inter- action inevitably alters the state of the quantum system, which causes loss of information encoded in the system. The system under consideration is not a closed system any more when interaction with the outside world is in ac- tion.

We formulate the theory of an open quantum system in this chapter by regarding the combined system of the quantum system and its environment as a closed system and subsequently trace out the environmental degrees of freedom. Decoherence is a process in which environment causes various changes in the quantum system, which manifests itself as undesirable noise. We will closely follow [1] and [2]. We deal with general quantum states described by density matrices.

We are interested in a general evolution of a quantum system, which is described by a powerful tool called a quantum operation. One of the simplest quantum operations is a unitary time evolution of a closed system. We assume the system-environment interaction is weak enough so that this separation into the system and its environment makes sense.

To avoid confusion, we often call the system of interest a principal system. The condition of weak system-environment interaction may be lifted in some cases. Let us consider a qubit propagating through a noisy quantum channel, for example. We have already noted that the unitary time evolution is an example of a quantum operation.

Other quantum operations include state change associated with measurement and state change due to noise. The latter quantum map is our primary interest in this chapter. Details of the environment dynamics are made irrelevant at this stage. A general quantum map does not necessarily satisfy these properties [3]. At this stage, it turns out to be useful to relax the condition that U t be a time evolution operator.

Instead, we assume U to be any operator including an arbitrary unitary gate. Let us consider a two-qubit system on which the CNOT gate acts. Suppose the principal system is the control qubit while the environment is the target qubit.

The unitarity condition may be relaxed when measurements are included as quantum operations, for example. Tracing out the extra degrees of freedom makes it impossible to invert a quantum operation. Therefore even though it is possible to compose two quantum operations, the set of quantum operations is not a group but merely a semigroup.

Here we give an interpretation of OSR as a noisy quantum channel. A quantum operation maps a density matrix to another density matrix linearly. What is meant here is a linear operator, acting on the vector space of Hermitian matrices, also acts on the space of density matrices and maps a density matrix to another density matrix. We will see further examples of quantum operations in this section and the next. This section deals with measurements as quantum operations.

This process happens with a prob- ability p i. However, it should be noted that they are not a unique type of measurements. Here we will be concerned with the most general framework of measurement and show that it is a quantum operation. Decoherence appears as an error in quantum information processing. The choice of the second qubit input state is far from unique and so is the choice of the circuit. Then the output of the circuit in Fig.

Equation 9. Figure 9. The Bloch sphere shrinks along the y- and z-axes, which results in the ellipsoid shown in Fig. This process is called the phase relaxation process, or the T2 process in the context of NMR. The fact that there are four Kraus operators suggests that the environment Hilbert space is at least four-dimensional.

In fact, we can construct a quantum circuit model shown in Fig. It is a Fredkin gate with the bottom control bit. The gate is an inverted Fredkin gate, in which the control bit is the third qubit.

This process describes the T1 process in NMR, for example. This equation, albeit exact, is not closed and is of little use in actual applications. One of the most popular assumptions is the Markovian approximation. For example we may assume that the behavior of the system is Markovian, if we are interested in a time scale much longer than the environment correlation times.

Now we are ready to derive the Lindblad equation by employing these tools. By separating the summation over J and K in Eq. Then Eq. The operators LK are called the Lindblad operators, while Eq. This shows that the coherence decay process is slower than the amplitude damping process. The second example is the Bloch equation in NMR. Let us take the Hamiltonian 9. Barnum, M. Nielsen and B. Schumacher, Phys. A 57, Kondo, et al.

Lindblad, Commun. Gorini, A. Kossakowski and E. Sudarshan, J. To reduce such errors, we must build in some sort of error correcting mechanism in the algorithm. Before we introduce quantum error correcting codes, we have a brief look at the simplest version of error correcting code in classical bits.

To reduce channel errors, we may invoke the majority vote. Namely, we encode logical 0 by and 1 by , for example. Note that the summation of all the probabilities is 1 as it should be. The success probability p0 increases as p approaches 0, or alternatively, if we use more bits to encode 0 or 1.

Assume the received bits are decoded according to majority vote. This method cannot be applicable to qubits, however, due to the no-cloning theorem. We have to somehow think out the way to overcome this theorem. General references for this chapter are [1, 2, 3] and [4]. We closely follow Steane [2] here. If she is to transmit a serise of qubits, she sends them one by one and the following argument applies to each of the qubits. A quantum circuit which implements the encoding The circuit a belongs to Alice, while the circuits b , c and d belong to Bob.

Table The set of two bits is called the error syndrome, and it tells Bob in which physical qubit the error occurred during transmission. These features are common to all QECC. We list the results of other cases in Table Note that among eight possible states, there are exactly two states with the same ancilla state.

Does it mean this error extraction scheme does not work? Now let us compare the probabilities associated with the same ancillary state. Observe that the intersection between thick solid lines and thick broken lines is an empty set. It is instructive to visualize what errors do to the encoded basis vectors as depicted in Fig. Therefore an action of a single error operator X can be corrected with no ambiguity. It should be clear from this observation that two-qubit encoding cannot correct single qubit error.

Now Bob measures his ancillary qubits and obtains two bits of classical information syndrome. Bob applies correcting procedure to the received state according to the error syndrome he has obtained. Suppose the syndrome is 01, for example. Figure This is nothing but the inverse transformation of the encoding It can be seen from Fig. The received state, which may be subject to an error, is then entangled with ancillary qubits which detect what kind of an error occurred during the state transmission.

In a sense, syndrome measurement singles out a particular error state which produces the observed syndrome. Once the syndrome is found, it is an easy task to transform the received state back to the original state.

Note that everything is done without knowing what the origial state is. Measurement of the error syndrome yields either 00 or We assume at most a single qubit is subject to the error.

To correct this error we have to identify at which qubit the error occurred during transmission. To this end, we introduce two ancilla qubits to each group of three qubits as shown in Fig. What is the syndrome A1 , B1 in each case? Let us check the syndrome A4 , B4. Each group of three qubits, after it goes through the Hadamard gates in the middle layer of Fig. Error syndrome has been extracted between two successive operations of the Hadamard gates.

What is the syndrome A4 , B4? Logical qubits may be implemented with fewer physical qubits as we show in the fol- lowing sections. We need to summarize classical error correcting codes before we introduce general theory of QECC. See [6] for more extensive accounts of the classical error correcting codes.

Our presentation closely follows [3, 4, 1] in this and the next sections. We assume the noisy transmission channel is symmetric, i.

The output Hct is called the syndrome. It is assumed that at most a single bit error occurs during transmission of a codeword. We need to identify which bit has erroneously transmitted, for which three bits are required. We assign for error-free transmission, for error in x1 , for error in x2 and so forth. It is found from the syndrome Bob obtained for c2 that x1 , x2 , x3 , x5 , x6 and x7 are received without error because at most a single error is assumed.

Bob applies a NOT gate on the fourth bit of the codeword he receives to recover the correct codeword. What is the syndrome he will obtain after applying the parity check matrix H.

Re- cover the correct code by making error correction. Repeat this for codes 1, 1, 0, 1, 0, 0, 0 and 1, 1, 0, 0, 1, 1, 1. This particular code generated by the above M is called the Ham- ming code. Note that M is obtained from H by adding the fourth row, with all the components unity.

The table for v and vM is v vM v vM Let us pick out an element of the code C, which contains even number of 1. Observe that they are mod 2 -orthogonal to all the members of C. The set of eight elements in Our choice of M as a generating matrix for the code is clear by now. A code which encodes k bits into a bit string codeword with length n and having the minimal distance d is denoted as n, k, d.

The Hamming code is thus characterized as 7, 4, 3. Therefore we can tell for sure which codeword is closest to the bit string containing an error see Fig. The original codeword is recovered by applying a NOT gate on an appropriate bit in the string. A single bit error sends the codeword to a point on a circle with the radius 1, which does not belong to the code C.

The minimal distance d must be at least 3 for single-error detection to make sense. We need to introduce three redundancy bits to attain the minimal distance of 3 in the code space C. Seven qubits must be required to encode a single logical qubit since there are more types of errors compared to the classical counterpart. Let us consider the encoding circuit for the coding Let us analyze the circuit shown in Fig. H stands for the Hadamard gate.

Collecting these facts, we construct the encoding circuit depicted in Fig. This fact sug- gests that we may use the set of these operators as the syndrome to identify the error occured in the received qubit.

This case corresponds to a two-qubit error Xi Zj , with which we will not be concerned. Now let us consider how they are measured. It is easy to see the measurement outcome A cor- responds to the eigenvalue of the operator Z1 Z2 , while B corresponds to that of Z1 Z3.

It turns out to be convenient to switch the control qubit and the target qubit by making use of the result of Exercise 4. Now the error syn- drome detection circuit in Fig. The above observation leads to the seven-qubit error detection circuit shown in Fig. We susupect that they might corresponds to the operators Mi. Accordingly the measurement of the ancilla reveals the eigenvalue of M0. Similarly other two gates evaluate the eigenvalues of M1 and M2.

Correction of the disrupted encoded state can be done by consulting with Eqs. An error operator Ek , which disrupts the code, is a tensor product of Pauli matrices, which anticommutes with some of Mi and Ni. This is done by applying gates in Fig. Let us look at a few examples. This desirable property of non-propagating error is called the fault tolerance, which is a key component in reliable quantum computing.

M4 is not an inde- pendent operator. Next, let us work out a quantum circuit which implements this encoding. Here UH1 is the Hadamard gate acting on qubit 1.

The error detection circuit is constructed following the strategy employed in the seve-qubit QECC. Correction of the disrupted encoded state can be done by consulting with Eq. The readout of the four ancillary qubits detects the syndrome. Shor, Phys. MacWilliams and N. Steane, Phys. Calderbank and P. Gottesman, Phys. Quantum Error Correcting Codes [10] A. Calderbank et al. DiVincenzo and P.

This new discipline called quantum information processing QIP is expected to solve a certain class of problems that current digital computers cannot solve in a practical time scale. Although a small- scale quantum information processor, such as quantum key distribution, is already available commercially, physical realization of large-scale quantum information processors is still beyond the scope of our currently available technology.

Classical information is encoded in a bit, which takes on values 0 and 1. Although a quantum computer with several qubits is already available for some physical systems, actual construction of a working quantum computer is still a challenging task. In the next section, we outline these conditions as well as two additional criteria for net- workability. The DiVincenzo criteria have been analyzed for several physical realizations, and the results of such analyses, as of the year , are sum- marized in [2].

We summarize the relevant parts of these criteria, which may be helpful in reading subsequent chapters in Part II.

A scalable physical system with well-characterized qubits. To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. A multi- qubit state is expanded in terms of the tensor products of these basis vectors.

Each qubit must be separately address- able. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional qutrit or, more generally, d-dimensional qudit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.

DiVincenzo Criteria Suppose you are not able to reset your classical computer. Then you will never trust the output of some computation even though processing is done correctly. Therefore initial- ization is an important part of both quantum and classical information processing. In many realizations, initialization may be done simply by cooling to put the system in its ground state. Alternatively, we may use projective measurement to project the system onto a desired state.

In some cases, we observe the system to be in an undesired state upon such measurement. For some realizations, such as liquid state NMR, however, it is impossible to cool the system down to extremely low tem- peratures. In those cases, we are forced to use a thermally populated state as an initial state. Long decoherence times, much longer than the gate operation time.

The hardware of a classical computer lasts long, on the order of 10 years. It lasts so long that we often have a problem giving up a healthy computer when the operating system is superseded by a new one. Decoherence is probably the hardest obstacle to building a viable quantum computer.

Decoherence means many aspects of quantum state degradation due to interactions of the system with the environment and sets the maximum time available for quantum computation.

This is not necessarily a big problem provided that the gate opera- tion time, determined by the Rabi oscillation period and the qubit-coupling strength, for example, is much shorter than the decoherence time.

A closed-loop control method incorporates quantum error cor- recting codes QECC introduced in Chapter 10, while an open-loop control method incorporates noiseless subsystem [4] and decoherence free subspace DFS [5]. Both of these methods, however, require extra qubits. Suppose you have a classical computer with a big memory. Now you have to manipulate the data encoded in the memory by applying various logic gates.

You must be able to apply arbitrary logic operations on the memory bits to carry out useful information processing. It is known that the NAND gate is universal, i. Note that a general unitary gate in U 2n is written as a product of an SU 2n gate and a physically irrel- evant U 1 -phase. This observation is noteworthy since the NMR Hamiltonian, for example, is traceless and is able to generate SU 2n matrices only.

Single-qubit gates are easily implemented if the one-qubit part of the Hamiltonian assumes two of the su 2 generators by properly choosing the control parameters, where su 2 stands for the Lie algebra of SU 2.

Implementation of a CNOT gate in any realization is consid- ered to be a milestone in this respect. Note, however, that any two-qubit gates, which are neither a tensor product of two one-qubit gates nor a SWAP gate, work as a component of a universal set of gates [7].

The result of classical computation must be displayed on a screen or printed on a sheet of paper to readout the result. Although the readout process in a classical computer is re- garded as too trivial a part of computation, it is a vital part in quantum computing.

The state after an execution of a quantum algorithm must be measured to extract the result of the computation. The measurement process depends heavily on the physical sys- tem under consideration. For most realizations, projective measurements are the primary method to extract the out- come of a computation. In liquid state NMR, in contrast, a projective measurement is impossible, and we have to re- sort to ensemble averaged measurements.

If this is the case, we have to repeat the same computation many times to achieve reasonably high reliability. Moreover, we should be able to send and store quantum information to construct a quantum data processing network. It may happen that some system has a Hamiltonian which is easily controllable and is advantageous in executing quantum algorithms. Com- pare this with a current digital computer, in which the CPU and the system memory are made of semiconductors while a hard disk drive is used as a mass storage device.

Therefore a working quantum computer may involve several kinds of qubits and we are forced to introduce distributed quantum computing. Interconverting ability is also important in long- distance quantum teleportation using quantum repeaters. Needless to say, this is an indispensable requirement for quan- tum communication such as quantum key distribution.

This condition is also important in distributed quantum computing mentioned above. The DiVincenzo criteria are not necessarily the gospel, and some condi- tions can be relaxed. For example, it is possible to replace unitary gates by irreversible non-unitary gates generated by measurements.

This idea is al- ready implemented in linear optics quantum computation [8]. Here is the list of some proposals: 1. Implementation of quantum error correction 3. Fault-tolerant quantum computing and 4. Topologically protected qubits. Low gate error rates 2. A way of remaining in, or returning to, the computational Hilbert space 4. A source of fresh initialized qubits during the computation and 5.

Benign error scaling: error rates that do not increase as the computer gets larger, and no large-scale correlated errors. Ability to perform gates between distant qubits 2. Fast and reliable measurement and classical computation 3. Little or no error correlation unless the registers are linked by a gate 4.

Very low error rates 5. High parallelism 6. An ample supply of extra qubits and 7. Even lower error rates. Many of the above conditions are necessary for quantum error corrections to work reasonably well. Here is the list of the candidates; 1.

Trapped ions 3. Neutral atoms in optical lattice 4. Cavity QED with atoms 5. Linear optics 6. Quantum dots spin-based, charge-based 7. Subsequent chapters in this book give detailed accounts of some of these re- alizations in the light of the DiVincenzo criteria. References [1] D. DiVincenzo, Fortschr. Nakahara, S. Kanemitsu, M. Salomaa and S. Takagi eds. Vartiainen et al. A 70, Knill, R. Viola, Phys. Zanardi, Phys. A 63, ; W. Ritter, Phys.

A 72, Palma, K. Suominen and A. Ekert, Proc. London A , ; L. Duan and G. Guo, Phys. Zanardi and M. Rasetti, Phys. Lidar, I. Chuang and K. Whaley, Phys. A 60, R ; D. Bacon, D. Lidar and K. A 60, Barenco et. DiVincenzo, Phys. A 51, Milburn, Nature , 46 Raussendorf and H. Briegel, Phys. See also D. Aharonov and M. Chen et al. In spite of its peculiar character associated with mixed states and lack of scalability, it still works as a prototypical quantum computer with at most 10 qubits.

We should point out that it is the only quantum computer commercially available at the time of writing this book. Molecules with a certain number of such nuclei are employed as a quantum register. Our exposition follows [1] and [2]. Other useful review is [3]. We restrict ourselves within liquid state NMR in the present chapter. Recently it has been recognized that NMR is also the most convenient system on which we can execute quantum al- gorithms. It is tempting to think that a spin 1 nucleus or a higher spin nucleus realizes a qutrit or a qudit more generally.

It is known, however, that these higher spin nuclei have very short decoherence times and are not suitable as computational resources. It also lists relevant references. TABLE Their molecular structures are shown in Fig. It is diluted in a solvent d-6 acetone. Nuclei working as qubits are in- dicated in boldface. The direction of B0 is taken as the z-axis throughout this chapter. The same coil is also used to pick up signals from rotating spins through magnetic induction when measurement is done.

This measurement is not a projective measurement, as was remarked in the beginning of this chapter. The measurement outcome is an ensemble average of an observable and is called an ensemble measure- ment. It should be noted that ensemble measurement is non-demolishing.

Moreover, quantum mechanically non-commuting variables may be measured simultaneously if ensemble measurement is employed. We can arrange several coils when operating several nuclear species simultaneously. Each coil produces rf pulses for a par- ticular nuclear species and receives induction signals from them. A sequence of pulses, namely a sequence of these parameters, is programmed beforehand according to a quantum algorithm to be executed and fed to the host computer in the beginning of an experiment.

The test tube and the coils in Fig. Due to rapid random motion of molecules in a liquid at room temperature, both rotational and translational intermolecular interactions are averaged to vanish, and each molecule may be regarded as being isolated from other molecules.

A typical value of B0 employed for quantum computing is on the order of 10 T. The metal cylinder in the right is a superconducting magnet generating B 0.

It also contains the test tube and the rf coils. The large box in the left is the spectrometer. Neumann, Math. Grundlagen der Quantenmechanik Dover, New York, , p. The mathematical tools of the quantum theory rely on the fundamentals of linear algebra —vectors and matrices of complex numbers. It may seem strange at first that we need to incorporate the machinery of linear algebra in order to Neumann , Math.

Grundlagen der Quantenmechanik Dover , New York , , p. That's because many quantum phenomena difficult to imagine concretely can be represented without further complications with some mathematical abstraction.

There are three basic concepts of mathematics - specifically linear algebra Part I The first part of the book provides the necessary foundations. This part starts with the definition of matter and rapidly takes the reader through the foundational concepts of quantum mechanics and the required math.

Krishnan Balasubramanian, Relativistic AD - FLD. In its initial form, Linear operators on infinite-dimensional function spaces are the basic objects of quantum mechanics. Skip to content Unlike more conventional treatments, this text postponesits discussion of the binary product concept until later chapters,thus allowing many important properties of the mappings to bederived without it. Author : K.

Author : George F. Author : C. Author : Mark M. Author : R.