Quantum computing pdf free download

C Write down the density operator. B Determine whether or not this is a pure state. Is this a mixed state? If not, why not? B If this is a valid density matrix, does it represent a pure state or a mixed state?

If mea- surements are made on systems in each of these states, what are the probabilities they are found to be in states 0 and state 1 , respectively? B Determine the density operator for the ensemble. D A measurement of Z is made on a member drawn from the ensemble. What are the proba- bilities it is found to be in state 0 and state 1 , respectively? B Compute the density matrix. C Find the density matrix that represents the reduced density operator as seen by Alice. D Show that the reduced density operator as seen by Alice is a completely mixed state.

C Does this matrix represent a valid density matrix? E Compute the components of the Bloch vector, and show that this is a mixed state. It is also important to focus on what the state of the system is after a measurement is made.

While measurement generally has no effect on a system in classical mechanics i. Much of the material in this chapter is a review of concepts already introduced. Measurement plays a fundamental role in quantum computation because, at some point, we have to be able to get information out of the computational system. In this chapter we will learn the basics about different measurement models used in quantum theory.

An act of measurement dis- turbs a quantum system in a fundamental way. After measurement the original state 6. The measurement of a quantum system involves some type of interaction or coupling of that system with a measuring device. That device can be thought of as part of the larger environment which the quantum system is a part of. Frequently the measuring apparatus or larger environment is known as the ancilla.

A system coupled to an environment is known as an open system. In Chapters 3 and 5 we discussed the time evolution of a quantum system. The systems considered in that case were closed quantum systems— that is, systems that were isolated from the larger environment. The dynamical behavior of a quantum system is determined by the Hamiltonian operator H, which describes the total energy of the system. However, the general solution of 6. We begin our detailed discussion of measurement by considering projective or Von Neumann measurements.

We have already introduced some basic notions of projective mea- surements in previous chapters, so some of this material will be review. The idea of making a projective measurement is based on the following notion: Given a set of mutually exclusive possible states, what state is the system is in?

We can use a projective measurement to determine if the atom is in the state g or in the state e. As another example, we may be interested in the position of a particle. Is it located at position x 1 or at position x 2? For a qubit, we could ask: Is the qubit 0 , or is it 1? Such mutually exclusive possibilities are described by projection operators in quantum measurement theory. If a set of orthogonal projection operators is complete, 6.

This is also another expression of the fact that probabilities must sum to one. The number of projection operators is determined by the dimension of the Hilbert space that describes the system.

However, the sum of two or more projection operators is, in general, not a projector. Result 6. When discussing pro- jective measurements, one frequently hears about the mysterious collapse of the wave function.

What this means is that while the state of the system prior to measurement could be a superposition of basis states as written in 6. These measurement results correspond to the basis states u 1 , u 2 , u 3 , u 4 , and u 5 , respectively. First we need to check and see if the state is normalized. What is the expectation or average value? Many applications in quantum computation involve composite systems. In this section we will go over some basic measurement operations on composite systems.

The utility of some of the relations stated in the previous section will become clear when dealing with the example of composite systems provided below.

Example 6. As we will see in the next chapter, this is a property of entangled systems. What is the state of the system after measurement? What is the probability that the system is found in the state if all 3 qubits are measured?

What is the postmeasurement state of the system? What is the expectation value of A? After this is done, what are the possible measure- ment results if both qubits are measured, and what are the respective probabilities of each measurement result? Following common notation, we denote a measurement operator by M m , where m is an index that denotes a possi- ble measurement result.

In the laboratory not all measurements are repeatable. The quintessential example is the detection of a photon— after it has been detected, the photon is destroyed. Hence repeated measurements on the system are not possible. A POVM is applicable in this case because it allows us to describe measurements on the system without regard to the postmeasurement state.

Describe a POVM that can distinguish between the two states. Assume that the states are normalized. These operators have provided a means of imperfectly distinguishing between two nonorthogonal quantum states. If the measurement outcome E 3 is obtained, no information about the state is available.

Moreover the wave function has collapsed in this case— the postmeasurement state of the system is 1. As we have seen, POVMs are a more general type of measurement that allow us to do things in quantum mechanics that are not possible using ordinary projective measurements. In later chapters we will see more applications of POVMs in quantum computation and information.

Let P1 and P2 be two projection operators. What is the average energy of the system? A measurement of X is made. B What is the probability that a measurement on the second qubit only gives 1? Show that the postmeasurement state is normalized. An X gate is applied to the second qubit. After this is done, what are the possible measurement results if both qubits are measured, and what are the respective probabilities of each measurement result?

Write down a POVM that allows for imperfect distinguishability between the two states. For the simplest two quantum systems case we denote the systems A and B. If these systems are entangled, this means that the values of certain properties of system A are correlated with the values that those properties will assume for system B.

The properties can become correlated even when the two systems are spatially separated—leading to the phrase spooky action at a distance. Quantum mechanics, however, tells a different story. For our qubit, say that we want to measure X. This is in direct contra- diction to the values held by EPR. EPR thought predictions such as this one for single systems by quantum mechan- ics were absurd enough. But things only get worse when you consider composite systems.

By making a very clever observation about what quantum theory tells us, EPR demonstrated that quantum mechanics predicts that if two particles interact and then separate, measurement of one of the particles will determine the values that the properties of the other particle must assume.

This is the case even though the particles are spatially separated and noninteracting at the time of measurement. In their brilliant paper, EPR focused on measurements of position and momentum. In discussions of this type it is common to denote the two quantum systems as being in the possession of Alice and Bob. We can even assume that they are so far away from each other that no signal—not even a light ray—can connect them over the time span when measurements are made.

However, prior to measurement, the values of each parameter, x A , x B or p A , p B , are not determined according to quantum mechanics. Alice could instead choose to measure the position of her particle. In principle, this is true even if Bob is on the other side of the galaxy.

We can summarize the conventional or classical view as local realism and the theories that are based on this philosophy as local realistic theories. Measurement of particle A in no way disturbs the state of spatially separated particle B. The values of measurable properties of each particle are objectively real. A simpler version of this thought experiment was put forward by David Bohm in and involved particles with correlated spins. The decay prod- ucts have to travel in opposite directions to conserve momentum, and their total spin has to remain zero in order to conserve angular momentum.

From 7. In this state there seem to be some strange correlations between the particles. If we only assume that conservation of angular momentum holds and say nothing about quantum mechanics, then the measurement results with respect to the Pauli operators Z , X will be found as listed in Table 7. As Table 7. To illuminate the difference between quantum mechanics and local realistic theories, we are going to have to consider a more complicated situation. When a system is entangled, this means that the individual component systems are really linked together as a single entity.

In summary, if two systems are entangled, the description of each system has to be made with reference to the state of the other system, even if the component systems are spatially separated and noninteracting.

This inequal- ity is experimentally testable, and to date all experimental evidence has come down in favor of quantum mechanics. We begin our thought experiment by imagining that an ensemble or large number of systems have been prepared so that Alice and Bob can measure spin along the three directions a, b, and c. Here we consider the local realist position, based only on conservation of angular momen- tum. The total number of particles is denoted by N , where there are N i particles found in state i as described in Table 7.

Again, assuming that Bob measures along a different direction, he will measure along b or along c. This gives you an idea of how the measurements work. The table shows that this occurs in populations N 1 and N 2. In each population there are six possible measurement results if we require Alice and Bob to measure along different axes. All the measurement possibilities are listed in Table 7. If all populations occur with equal frequency then half of the measurements will be such that Alice and Bob obtain opposite measurement results.

By using some basic math regarding real numbers, we can derive some inequal- ities for the data in Table 7. To see what quantum mechanics says about the situation, we need to consider a qubit oriented in an arbitrary direction. We prepare the system in the singlet state. A clear distinction has been made between simple counting arguments based on local realism Table 7. Experiment agrees with the predictions of quantum mechanics, so theories of the type that Einstein favored based on local realism are ruled out as descriptions of nature.

When a system consists of two subsystems we say it is a bipartite system. An example of this is when Alice and Bob each have one member of an entangled pair of particles. It can be expanded in terms of the basis states 7. For an example of a basis for a bipartite system, consider the Bell Basis. Example 7. When two systems are entangled, the state of each composite system can only be described with reference to the other state.

If two states are not entangled, we say that they are a product state or sepa- rable. Find the allowed energies of the system and show that the eigenvectors of the Hamiltonian include entangled states. Solution First we express the Hamiltonian in matrix form. Is this a separable state? We will denote the Schmidt number by Sch. Is this state separable? What is the Schmidt number? Derive the result 7. Does it have a similar form?

Consider the eigenvectors in Example 7. Use 7. Derive 7. Can the following state be written in diagonal form in terms of the Bell basis? We can move it from one place to another, or we can do some type of basic processing on the information using a logic gate.

Sets of logic gates can be connected together to construct digital circuits. In this chapter we will be introduced to the equivalent notions of logic gates and circuits in a quantum computer.

We begin with a brief overview of classical logic gates. A simple example is the NOT gate. The NOT gate is a single input gate.

That is, if the input to the gate is a 0, the output is a 1, and if the input is a 1, the output is a 0. This is done using a truth table, which is a table that lists the inputs together with the corresponding outputs of the gate. For a NOT gate, this is very easy to do.

We write the values of the single input bit on the left side of the table and the corresponding outputs on the right: Input NOT 0 1 1 0 Now we will consider more complicated operations that are applied to pairs of bits. The OR gate accepts two bits as input, which we label A and B. This gate works by inverting the result of the AND gate. That is, all computing operations can be completed using only NAND gates.

As a simple example of how other logic operations can be implemented using only NAND gates, suppose that happens if we supply the same bit to both inputs of the NAND gate. The NAND gate is interesting because it is universal, but it is also irreversible. That is, looking at the output of a NAND gate, we cannot work backward to deter- mine the values of the input bits once the gate has acted upon them.

However, it turns out that there are reversible gates that can be used to construct a classical computer. We denote the control bit by C; its function is to determine whether or not a given operation will be applied to the other input bits.

Recall that quantum operators can be represented by matrices. A quantum gate with n inputs and outputs can be represented by a matrix of degree 2n. Following the procedure used when thinking about classical logic gates, we begin by examining the simplest gate possible, the quantum NOT gate. Example 8. Use the matrix and outer product representations. There are other special cases of interest. Solution The matrix representation given in 8. We can regroup the terms in 8.

In fact we can create rotation operators to represent rotation about the x , y, and z axes on the Bloch sphere by exponentiating the Pauli matrices. The rotation matrices given in 8. For example for the T gate of 8. For example, the representations of the Pauli operators X , Y , and Z and their action on a single qubit are shown in Figure 8. The circuit diagram representation of a Hadamard gate is shown in Figure 8. We represent measurement in a circuit with an encircled M. This is shown in Figure 8.

In other texts or papers the representation of measurement may be different, but it should be clear from the context. We will see how to incorporate measurement into quantum circuits in later chapters, for example, when we consider teleportation in Chapter In this section the notion of a controlled gate will allow us to implement an if —else type of construct with a quantum gate.

Consider a controlled classical gate. We include a control bit C. When working with two-qubit gates, we consider their action with respect to two-qubit states.

If the control qubit is 1 , then the NOT or X matrix is applied to the target cubit. To write the matrix representation of the controlled NOT gate, we have to do it with respect to the states 00 , 01 , 10 , and Using 8. From 8. Of course, if we look back at 8. So we will start with a qubit given by 0 and act on it with a Hadamard gate.

We can use a similar thought process to see how to generate the other Bell states. The circuit required to do this, in general, is shown in Figure 8.

One thing to notice about this circuit is that moving from left to right indicates the passage of time. So a wire in a quantum a H b bab Figure 8. Time moves from left to right, with wires used to represent the passage of time where the state is left alone. First we apply a Hadamard gate to the qubit —a to generate a superposition state.

This is then used as the control bit in a CN gate. As we mentioned earlier, it is possible to generate any type of controlled U gate we wish. For example, we can have a controlled-Hadamard gate. The action of the controlled Hadamard gate is as follows: If the control qubit is 0 , nothing happens to the target qubit. If the control qubit is 1 , then we apply a Hadamard gate to the target qubit. Solution Using the matrix representation of the CH gate, we need to write out the states 01 and From 4.

Then consider the case where the target qubit is 0. This procedure is illustrated schematically in Figure 8. An equivalent circuit, consisting of two controlled NOT gates and the single-qubit gates A, B, and C , will result in the same output. To illustrate how circuits can be manipulated, we have the common example illustrated in Figure 8. We want to prove that a controlled NOT gate can be written in terms of two Hadamard gates and a controlled Z gate. So we start here and use 8. Describe the action of the Y gate in terms of the Bloch sphere picture.

B Describe the action of the Hubbard operators on the Hadamard basis states. Show that the controlled NOT gate is Hermitian and unitary. Write down the matrix representation for the controlled Z gate. Then write down its representation using Dirac notation. Find the square of each operator. A large part of the desire to develop a quantum computer has come from the discovery that some algorithms work dramatically better on a quantum computer than they could ever work on a classical computer.

This is because the nature of quantum systems— captured in superposition and interference of qubits— often allows a quantum system to compute in a parallel way that is not possible even, in principle, with a classical computer. It was discovered that given a function f x , a quantum algorithm is capable of evaluating the function at multiple values of x simultaneously. As we will see below, a quantum algorithm highlights one of the central tugs-of-war that exist in quantum theory.

A qubit can exist in a superposition of states, giving a quantum computer a hidden realm where exponential computations are possible. In other words, the fact that a quantum system can exist in a superposition or linear combi- nation of states allows us to do simultaneous parallel computations that cannot be done even, in principle, on any classical computer.

This feature allows a quantum computer to do parallel computations using a single circuit-providing a dramatic speedup in many cases. Nature allows us to get around this to a certain extent and extract useful infor- mation using quantum interference. This is another feature of qubits that cannot be seen with classical bits— and it plays an important role in the development of useful algorithms.

At the present time only a small set of truly quantum algorithms have been discovered, and this remains a very active area of research. We begin by reviewing two quantum gates that are important in the develop- ment of quantum algorithms and explaining quantum interference.

Now consider what happens when we apply two Hadamard gates in parallel. When n Hadamard gates act in parallel on n qubits, this is called a Hadamard transform. So the operation shown in 9. By summing over the variable x , we can write the states compactly. For relation 9. Example 9. There are two basic rules that can be followed.

We represent a set of operations performed in series i. This is illustrated in Figure 9. The matrix representation of this sequence of operations is written down by multiplying the matrices in reverse order. Hence the operation shown in Figure 9. This is a manifestation of quantum interference—mathematically this means the addition of probability amplitudes. There are two types of interference, positive interference inwhich probability amplitudes add constructively to increase or negative interfer- ence in which probability amplitudes add destructively to decrease.

In the case of 9. That is, interference allows us to deduce certain global properties of the function. Earlier we mentioned that quantum parallelism can be described as the ability to evaluate the function f x at many values of x simultaneously. So a function on a single bit can be constant or balanced. Whether a function on a single bit is constant or balanced is a global property. Now, since x is a qubit, it can be in a superposition state.

Using algebra, we write the action of the circuit shown in Figure 9. Suppose that f x is the identity function. But remember how quantum measurement works. If we measure the state x f x , we get one and only one value of x and f x. Moreover the value of x that we obtain is completely random. So what we have so far is the same as evaluating the function f x at some randomly chosen x.

For a simple function on bits we can learn the value of f 0 or f 1 , but not both simultaneously even though they are simultaneously present in the premeasurement state. Apply Hadamard gates to the input state 0 1 to produce a product state of two superpositions. Apply U f to that product state. We calculated it in 9.

Again, this algorithm allows us to determine whether a function f x is constant or balanced, but this time the function has multiple input values. If f x is constant, then the output is the same for all input values x.

We start with an initial state that includes n qubits in the state 0 and a singlequbit in the state 1. Hadamard gates are applied to all qubits. The circuit is illustrated in Figure 9. It might not be immediately obvious looking at 9. In this case f x is constant. This shows that the function is balanced. Notice that we have two things going on here—superposition states and the introduction of phases.

This is how we get the form 9. The discussion so far has been pretty abstract. The circuit used to do this is shown in Figure 9. The control bit for this gate is the state x 1 , which is 0 or 1. The eigenvalues of unitary operators are phases. Acting with m Hadamard gates on 9. Notice that the state in 9. From 9. Doing so involves looking at the terms in 9.

We drawing a unit circle in the complex plane. As illustrated in Figure 9. In Figure 9. This is an algorithm that can be used to factor prime numbers—meaning that it can be used to break encryption codes if a practical quantum computer is ever built.

Needless to say, this algorithm got the attention of a lot of people. This is where we stop. With large numbers it will swamp the best computers available, the time required is exponential in log N. To obtain the exact value, continued fractions are applied. The presentation given here is simplistic and is not something that has much real world utility.

Now suppose that the bit string is small, just 5 bits. Next, using 9. It does so only a tiny bit at a time, so we have to apply it multiple times, say, m times. In that case 9. Derive 9. Quantum gates are universal in the sense that quantum gates can be designed that do anything a classical gate can do. A qudit system is more complicated but might be considered because of the increased computing power. Otherwise, it is balanced. Derive the relation 9. Consider the eigenvectors 9.

This is because we can use entanglement to accomplish communications and information processing tasks that would otherwise not be possible. In this chapter we will explore two areas where entanglement can be used to do some rather unusual tasks. By using entanglement, Alice and Bob can set up a quantum communications channel that links them together in a quantum way via the EPR paradox—allowing Alice to send her state to Bob in an almost magical fashion.

In our second application of entanglement, we will look at superdense coding. This is a procedure that allows us to send two classical bits to a party using only a single qubit— demonstrating the power of quantum information processing.

The basic idea is that you get scanned somehow, turned into energy, then beamed to where you want to go and rematerialized. It allows us to send a quantum state from one place to another without that state traversing the space in between. While teleportation seems to work almost by magic Einstein can breathe a sigh of relief because special relativity seems to step in to prevent faster than light communication.

The task at hand is that Alice wants to transmit an unknown quantum state to Bob. Teleportation takes place in a series of steps. We begin by creating an entangled EPR pair. Now Alice and Bob physically separate. Alice decides that she wants to send the state She can do it by letting it interact with her member of the EPR pair in So indicates that Alice has a 01 in her possession while Bob has a 1. Alice begins interacting her member of the EPR pair, which is the second qubit in Teleportation Step 4: Alice Measures Her Pair The next step in the process is that Alice makes a measurement on both qubits in her possession.

So we can write The answer is Alice gives him a call. Alice has to somehow tell Bob her measurement result, and she has to do it using a classical communications channel—a telephone, email message, radio wave, or something— some mechanism governed by the speed of light limit.

But security is maintained—Alice just calls Bob and for instances, says she got 01, then Bob applies his X gate to obtain the state Alice wanted to send to Bob. Nothing about that state is communicated over the classical channel—Bob can obtain it because they shared an entangled EPR pair of particles.

The lesson here is that quantum information based communication can be characterized by two key aspects—local operations and classical communications LOCC. If classical communications is not used, then the state will appear totally random to Bob. For us, we can use it as a vehicle to learn more tools in the toolbox of the quantum information theorist. We begin by considering the Peres partial transposition condition, which allows us to determine whether or not a given density operator represents an entangled state.

If the eigenvalues are all positive, then the state is separable. Example Show this using the Peres partial transposition condition. The presence of the negative eigenvalue 2 tells us that this is an entangled state. Returning to teleportation, it is possible for Alice to simultaneously transmit two unknown quantum states to two parties, Bob and Charlie.

To simultaneously teleport the states The state Based on her result, which she communicates with a classical channel, Bob and Charlie perform local unitary oper- ations using the Pauli matrices I , X , Y , and Z.

However, they do not yet have the states they need. Solution We begin by writing down the density matrix of the system prior to locking. These are the second and fourth qubits A2 , C. The state after locking is given by There are 64 terms in the expansion, you will have to take my word for it. Therefore the locking operation has destroyed the entanglement between Alice and Bob. This is possible, in principle, even if the particles are light years apart.

Entanglement swapping begins with two EPR pairs. We label the qubits 1, 2, 3, and 4. Alice has qubits 1 and 4 in her possession while Bob has qubits 2 and 3 in his possession. Now Alice performs a Bell state measurement on particles 1 and 4.

Now particles 2 and 3 are entangled. Alice and Bob create two entangled states. Particles 1 and 2 are entangled together, and particles 3 and 4 are entangled together. Alice has particles 1 and 4 in her possession, and Bob has particles 2 and 3 in his possession. What is teleported here? Perhaps you can say that the entanglement has been teleported. This procedure works even if particles 2 and 3 have never interacted.

Alice does her Bell state measurement—now Bob and Charlie share an entangled pair and can use it as a resource between the two of them—so Bob can teleport a state to Charlie.

This time Alice would like to send Bob two classical bits of information, but she only has one single qubit to work with. She can accomplish this amazing feat using a protocol known as superdense coding.

Once again, we begin with Alice and Bob sharing an entangled pair of particles. Alice transmits the information to Bob, this time by actually sending Bob her qubit. If she wants to send Bob the classical bit string 00, then Alice leaves her qubit alone.

Then the state stays in its initial state So he obtains one of the classical two-bit strings 00, 01, 10, or Show that if Alice takes qubit 1 and Bob takes qubits 2 and 3, Alice can perform a local unitary operation and send Bob her qubit, which results in the transmission of two classical bits to Bob. Now suppose that Alice applies the X gate.

What does Bob see? Start with the state You should see that Bob sees a set of states that are all equally likely—so Bob sees a completely random mixture.

Show that this scenario can be used for superdense coding. Later the key can be used to recover or decrypt the message. Currently cryptographic keys are generated using mathematical algorithms that are hard but not impossible to break.

In contrast, quantum cryptography is a method of key distribution that relies on the laws of physics to create a key. Suppose that two parties, denoted Alice and Bob as usual, want to share a private message. A trivial way to encrypt the message is to generate a key k that is just a number used to scramble up the message.

We can add k to each character in the message, for example, converting a useful message into a scrambled bunch of meaningless characters. So how can Alice and Bob secure the message?

Then the scrambled message is sent over the public communications channel. When it reaches Bob, he subtracts k to decrypt or recover the original message. We add this to each character in a given message. Eve taps the line and gets the meaningless string of characters EDG and has no idea what Alice is talking about. We can minimize the risk by changing the key, maybe we change it every time we send a message, say.

When we regularly change the key we are using a one-time pad method. More safeguards are needed to ensure the security of our data, and one method that is very popular is an encryption scheme called RSA. The basic idea behind RSA is to use two keys, one that is public and one that is private. You would need to factor the numbers in order to crack the system. We begin by choosing two prime numbers, which we denote p and q. A prime number is a natural number p that is divisible only by itself and by 1.

For example, some small prime numbers are 2, 3, 5, 7, 11, 19 The number 9 is not a prime number because it is divisible by 9, 3, 1. The number 13 is a prime number because its only divisible by 13 and 1. We generate the public key using n and e. Notice that if we knew d , since we know e which is available on a public channel , we would be able to decrypt the message.

Is there a better way to encrypt messages? It turns out that quantum mechanics shows us several ways. This is known as quantum key distribution, or QKD. This includes an ordinary public channel, which is just an ordinary classical communications link—it could be the Internet, a cell phone, or your home telephone. Encrypted messages are sent over this line. In addition a second piece of the QKD puzzle is used—a quantum communications channel over which the quantum key is distributed.

In practice, this is done using individual photons in different polarization states. Quantum mechanics relies on a fundamental principle of quantum theory—that measurement disturbs a quantum state. To learn something about a key encoded as a quantum state, a measurement has to be made.

So if Eve taps into the line, she has to make measurements—disturbing the system in such a way that Alice and Bob can detect her presence. The no-cloning theorem—quantum states cannot be copied. As a result Eve cannot tap a quantum communications channel, copy the quantum states used to create the key for herself, and send the originals down the line to Bob.

Measurement leads to state collapse. A key aspect of QKD is that different bases will be used to create a bit string. When we make measurements in one of the given bases, we will cause state collapse such that measurements in the other basis are completely random. In other words, extracting some information about the state disturbs the state of the system. Measurements are irreversible. Suppose we obtain measurement result 0.

Alice then sends the random sequence of qubits over the quantum channel to Bob. We have two bases here, and Alice randomly selects what basis to use to create each bit in the string.

Alice and Bob compare notes, only telling each other which basis was used at each position. When Alice and Bob use a different basis, they discard that qubit. We call the key that results after discarding the bits where Alice and Bob used different bases the sifted key. Solution We simply throw out the bits where Alice and Bob used a different basis, numbering the bits from 1 to 8.

If the error rate is too high, this probably indicates Eve is listening in. In Example Suppose that she selected the computational basis. If we include a large number of qubits in our string, Alice and Bob can use this kind of behavior to deduce the presence of Eve. In our example Alice knows she created the bit string But Bob has the bit string Noise on the quantum channel will also create errors.

If Alice gets 1, Eve has 1. This time Alice and Bob use two nonorthogonal bases. The basic procedure is the following: 1. Bob uses a public channel to tell Alice what bit positions he has. The way this works is that if Alice creates a 0 , and Bob measures in the compu- tational basis, he obtains 0.

Notice the differ- ences when comparing this to the BB84 protocol. In BB84, Alice generates a key. Alice and Bob measure their respective qubits in randomly chosen bases. They keep these bits to create the key. Since the measurement results will be perfectly correlated or perfectly anticor- related, it is easy for Alice and Bob to determine whether or not an eavesdropper is present. Ordinary errors can be corrected using quantum error correction techniques.

Basically all this involves is throwing away qubits that Eve could have attempted to measure. These are quantum systems that do not interact with the outside world. That is, an idealized model. In reality, quantum systems interact with the outside environment. The problem if that interactions with the environment can introduce noise and cause errors. To deal with this and construct, for example, real quantum computers and communications systems, we are going to need some kind of error correction.

Before we get there, we are going to have to develop a mathematical formalism to describe quantum systems that interact with the environment. We refer to systems of this type as open systems.

Open quantum systems are important for the following reason: in an open quantum system, a pure state can evolve into a mixed state. The downside of this is that we need pure states to do quantum computation, and hence this type of evolution into mixed states is undesirable. In this chapter we will describe some of the formalism used to describe open quantum systems and then discuss error correction techniques. We have seen through- out the book that the ability to work with superposition states Unfortunately, when a quantum system interacts with the environment we often say the system is coupled to the environment , superposition can be lost.

We call this process—whereby a pure state is turned into a mixed state via interactions with the environment—decoherence and refer to states like The idea behind coherence is that the amplitudes in To see this, we apply a Hadamard gate to the qubit. What happens in the case of mixed states? If a state is mixed an incoherent mixture; see Chapter 5 , the amplitudes cannot interfere. The amplitudes did not interfere in this case.

The identity operator, of course, represents no error at all. We can summarize the effect of noise on a single qubit by saying that quantum noise acts on qubits via the application of one of the operators I , X , Y , Z. Which operator is applied depends on the state of the environment. What we are imagining now is the interaction of the principal system with the environment, during which the opera- tion U is applied.

We can do this by tracing out over the environment. Note that the Kraus operators act on the principal system. Find the Kraus operators and write down the operator-sum representation as a matrix.

Solution Since the environment is a single qubit, the basis for the environment is the computational basis. Hence we just let A pass through and only consider the action of B when deriving the Kraus operator. Find the density operator of the principal system after the interaction. Solution Recall that the controlled NOT gate acts on two qubits. If the control qubit is 0 , then nothing happens to the target qubit.

The basic idea is that Alice transmits a qubit to Bob. This is done through some communications channel with noise or distortion. This is known as the depolarization channel. What are the Kraus operators that can be used for the operator-sum representation of the depolarization channel?

Expanding out When describing a quantum system undergoing energy dissipation because of some type of interaction with the environment, we apply a quantum operation known as amplitude damping. Amplitude damping describes a decay process. The example we will use is an atom decaying from an excited energy state 1A to the ground state 0A. On the other hand, This can be accomplished using On the other hand, the environment goes from the ground state to the excited state, indicating the need to use Therefore 0 is the lowest energy or ground state of the system.

In the next problem we consider a quantum system a interacting with an environment r where both the system and the environment are harmonic oscillators for example, the environment could be a heat reservoir. This is a more complicated example of amplitude damping. Let the number states of the system be denoted by n a and the states of the environment by n r.

So our goal is to derive an expression that gives the k th Kraus operator. Using this expression together with During phase damping the principal quantum system becomes entangled with the environ- ment. Of course, this is undesirable if we are trying to use the quantum system to perform quantum computing.

We assume that the environment is initially in the state 0E. All information about the relative phases in the original state of the principal system is lost. This is phase damping. If the system is in the state 0 , the environment is unchanged. Algorithms that can be appropriately.

Just Now communication, quantum cryptography, and quantum computing. It is seen that the richness of quantum physics will greatly a ect the future generation technologies in many aspects. Posted in : Intellectual Property Law Show details.

Just Now Quantum Computing : Progress and Prospects provides an introduction to the field, including the unique characteristics and constraints of the technology, and assesses the feasibility and implications of creating a functional quantum computer capable of addressing real-world problems.

This report considers hardware and software requirements. All you need to do is just click on the download link and get it. If you liked it then please share it or if you want to ask anything then please hit comment button. Quantum — The smallest amount or unit of something, especially energy. At present, we do not have a quantum RAM qRAM capable of efficiently encoding this information as a quantum state, and reliably storing it for extended periods of time.

July 19, While many attorneys may already be familiar with the gains new innovations like blockchain and artificial intelligence offer, few have even heard of the biggest and most promising technology on the horizon — quantum computing. As technology plays an increasing role in the practice of. Up to now most of our work has been at a very low level, seeming more an exercise in linear algebra than a discussion of a new model of computing.

Currently, it seems that similar statement can be made for quantum annealers in term of number of qubits D-Wave. Maybe, also for universal quantum gate-based computers. That paradigm, however, was justified for the timing relations of vacuum tubes only.

The technological development invalidated the classic paradigm but not the model! It led to catastrophic performance losses in computing systems, from the operating gate level to large. Quantum computing is the practice of studying quantum computers and their potential. Posted in : Study Law Show details. August 13, Comment 1. In classic computing , uncertainty is unacceptable.

Quantum computers have an innate ability to learn about the world, dealing in probability, as they explore multiple answers to come up with complex. The company has operated quantum computers over the cloud since and serves global enterprise, government, and research clients through its Rigetti Quantum Cloud Services platform. The tech titan said its new IBM Quantum System Two, announced on Monday, is its first step in its data-center-style approach to quantum computers, and an acknowledgment that quantum computers aren't all alike.

As predicted in by Gordon Moore, the co-founder of Intel, computer processing power has more than doubled every 24 months. The cost of processing power has seen a 10 billion-times decrease in the first years of the computer age, 50 …. Posted in : Sea Law Show details. You can change your mind at any time by clicking the unsubscribe link in the footer of any email you receive from us, or by contacting us at [email protected].

Posted in : Consumer Law Show details. International Journal of Applied Systemic Studies. International Journal Computational Vision and Robotics. International Journal of Modelling, Identification and Control. International Journal of Power and Energy Conversion. The last such problem I attacked had 40, constraints and , variables. I found a feasible solution within 0. Quantum computing principles use the principle of coherent superposition storage.

A quantum computer thus has the theoretical capability of simulating any finite physical system and may even hold the key to creating an artificially intelligent computer. Many problems in finance can be expressed as optimization problems.

These are tasks which are particularly hard for classical computers, but find a natural formulation using quantum optimization methods [7]. In recent years, this field has known a tremendous growth, partly due to the commercial availability of quantum annealers.

Nielsen and Isaac L. Chuang Readings: posted online with the syllabus for each lecture. These are critical. Introduction Linear Algebra Quantum Mechanics. While interestingly, they are not universally faster than classical computers, they do perform specific types of calculations faster. Quantum Computing Pdf Free faqlaw. Quantum Computing Basics Pdf faqlaw. An Introduction to Quantum Computing 1 hours ago computing model.

In this book we examine File Size: 2MB. Quantum Computation: a Tutorial 9 hours ago second part20 will be concerned with programmatic perspective on quantum computation. Short introduction to Quantum Computing Kattemolle 4 hours ago 3 Quantum computers A quantum computer works with quantum bits qubits. Quantum Computing Lecture Notes 5 hours ago quantum computing.