QI Chapter 1

Quantum Computation and Quantum information

Due to the relationship with the rapidly developing IT industry, quantum computing and the development of quantum information have attracted a lot of capital attention in recent years, with Microsoft at the University of California, Santa Barbara, which has a sation Q lab, focusing on topological quantum computing, the Martinis of quantum computing. The lab led by Google acquisition and so on. People are imagining the great changes that quantum computers can bring to our society after they are put into use, and the "quantum hegemony" is increasingly being mentioned, and there are almost constant laboratories around the world claiming that they have achieved more quantum entanglement, as if quantum computers are already within reach.

But as a member of the quantum computing industry, what I know about quantum computing is quite a long way from the public's good expectations. I hope that we can briefly introduce the basic principles of quantum computing and quantum information, as well as supervise our own study and master the basic theory, and prepare for future scientific research, based on the main bibliography: M. A. Nielsen, I. L. Chuang, Quantum Computation and quantum information.

Chapter 1. The quantum mechanics involved

The theoretical framework of quantum mechanics is the Hilbert space, which is very familiar to the students of mathematics majors, but for physics, it is possible to discard some of the rigorous definitions and cumbersome proofs, after all, the correctness of the theory is not determined by us, but by nature itself. In many of the textbooks of quantum mechanics, it is possible to construct quantum mechanics at the beginning by introducing many assumptions as axioms, but it is hard to do so, and there will be textbooks that obscure some simple theories, but such textbooks are filled with classrooms in many universities ' physics departments. Dirac, one of the founders, wrote in 1936 that the fundamentals of quantum mechanics are much higher than many of today's books.

Section 1. State superposition principle

We use quantum states to describe particles, writing \ (|\psi\rangle\) , which is introduced by Dirac and similar symbols \ (\langle\psi|\) , Dirac is called ket and bra respectively, from parentheses bracket. If a moment we can also use \ (|\phi\rangle\) to describe this particle, then the state \ (c_1|\psi\ Rangle+c_2|\phi\rangle\) is also the state in which the particle is located. Here we introduce the addition and multiplication of states, \ (c_1\) and \ (c_2\) for any complex number, So we find that the state is actually a vector in the linear space, and we have given its addition and multiplication, which is all the possible states of the particle, the Hilbert space. At the same time, we also need to define the inner product of any two states, in order to give the length of the gauge, two states to finish the inner product is a complex number. For the state \ (|\psi\rangle\) and \ (|\phi\rangle\) , the inner product can be written as span class= "Math inline" >\ (\langle\phi|\psi\rangle\) or \ (\langle\psi|\phi\rangle\ , where \ (\langle\phi|\psi\rangle^*=\langle\psi|\phi\rangle\) , This method of two states to denote the inner product is the inspiration Dirac the word bracket split the source, of course ket must and only and bra do the inner product, each KET can find a bra, only need to simply put the right half parenthesis into the left side bracket, for beginners do not need to understand the rigor behind The definition.

Section 2. Measurement

When we try to use some instruments to measure the momentum or the coordinates of a particle, it has quietly changed its state, and it is not obvious to macroscopic objects, but the effect in the microcosm is significant. So we consider the magnitude of the momentum or the coordinates as the operator, and we will get a new state in the state. We know that there are many operators in linear space, but the operators that can be measured as physical quantities must be Hermitian (Hermite), that is, the conjugate transpose equals itself, which derives from the two excellent properties of the matrix. First, the eigenvalues of the Hermitian matrix are real numbers, and secondly, the eigenvector orthogonal to the different eigenvalues of the Hermitian matrix. As a result, quantum mechanics thinks that each time we measure, we get a random eigenvalue of the Hermitian matrix, which is consistent with real life experience, because the coordinates or momentum is obviously a real number. So how do you calculate the probability of getting a eigenvalues \ (a\) ? Based on the second property of the Hermitian matrix, we can use these orthogonal eigenvalues as a set of bases in the Hilbert space, and if we measure the state of the particle as \ (|\psi\rangle\), the eigenvalues \ (a\) correspond to the intrinsic state of \ (|a\rangle\), the probability is \ (|\langle a|\psi\rangle|^2\), the state is projected to the mold side of the eigen state. It is possible to verify that this is self-consistent because of the probability and \ (\sum_a|\langle a|\psi\rangle|^2=1\), which can be obtained by all the eigen-states orthogonal and complete.

Here is not the end, for a particle, after the measurement it will randomly become the operator of a certain eigenvalues, the measured value is the intrinsic value, we imagine if there are a large number of identical particles, and the same operation, and then the results are averaged by probability, that is, the average value of the measurement, is recorded as \ (\ langle\psi| a|\psi\rangle\), where a is the operator to be measured. In most cases, the average is more important, and the actual measurements we get are often averages.

With the concept of an average, we introduce the concept of variance, as in mathematics, to define the variance or error of measurement A:
\[(\delta A) ^2=\langle\psi| (A^2-\langle a\rangle^2|\psi\rangle\]
The Heisenberg uncertainty relationship can be expressed as:
\[\delta X\cdot\delta P\ge \frac{\hbar}{2}.\]
Proving the need for another fundamental assumption of quantum mechanics, namely:
\[[x,p]=i\hbar.\]
The parentheses in the above equation are for the a,b]=ab-ba\, i.e. \ ([), combined with the Schwartz inequality, can prove the Heisenberg uncertainty relation.

Section 3. Direct Product Hilbert space

At present we can only deal with a single particle, which is obviously not enough, while processing multiple particles is the focus of the study, it is necessary to combine the Hilbert space of each particle to get a larger Hilbert space, this operation is the direct product. The new state space in which particles \ (i, I=1, 2, ..., n\) are in common is \ (\otimes_ih^i\), which is very similar to the Cartesian product. Operators in the new space also need to be concatenated with a straight product symbol, defined as:
\[a\otimes B (|\phi\rangle\otimes|\psi\rangle) = (a|\phi\rangle) \otimes (b|\psi\rangle). \]

Section 4. Density operator

We extend the definition of the above state further. In practice, we do not know which state the particle is in, only that it may have half of its concept in a certain state, and half the probability is in another state, which we call the mixed state. To describe this state, we use the concept of the density operator proposed by Landau. Let's look at the average of the measurements in this state, which is obviously a heavier average than the definition just now. Suppose the particles are in state\ (|\psi_i\rangle\)On the probability of\ (p_i\), the average of the measurement operator A is:
\[\langle A \rangle=\sum_ip_i\langle\psi_i| A|\psi_i\rangle=\sum_{i,a}p_i\langle\psi_{i}|a\rangle\langle a| A|\psi_i\rangle=\sum_{i,a}\langle a| A|\psi_i\rangle p_i\langle\psi_i|a\rangle=tr (A\rho), \]
which
\[\rho=p_i|\psi_i\rangle\langle\psi_i|\]
is the density operator. In the above calculation we take advantage of the completeness of the eigen state of the Hermitian matrix, and note that similar to\ (|\psi_i\rangle\langle\psi_i|\)is written as a projection operator, which is easy to understand. It can be found that, as defined above, there are three properties of the density operator, semi-positive qualitative, hermite sex and\ (tr (\rho) =1\)and is easy to verify when and only if\ (\rho^2=\rho\), the state in which the system is located is pure state. If we have a two-particle density operator, how do we calculate the average of a physical quantity of one of the particles? Let's consider the state of these two particles as pure state.\ (|\psi\rangle\), then the average value of the particle 1 measurement of the mechanical quantity A is:
\[\langle\psi| A\otimes i|\psi\rangle=tr_{12} (|\psi\rangle\langle\psi| A\otimes I) =tr_1 (tr_2 (|\psi\rangle\langle\psi|) A) =tr_1 (\rho_1a), \\rho_1:=tr_2 (|\psi\rangle\langle\psi|). \]
In the above formula we define a new density operator, and the resulting new density operator is called the approximate density operator. It is important to note that Particle 1 's reduction density operator is a trace of particle 2, that is, using the trace operation to "remove" the part of the particle 2 in the state.

Section 5. Schmidt decomposition

Schmidt decomposition is so important in quantum computing that it needs to be explained in detail in a single section, which is extremely important when dealing with mixed states. The basic idea is to write the mixed-state density operator:
\[\sum_ip_i|\psi_i\rangle\langle\psi_i|=\sum_{jk}a_{jk}|j\rangle|k\rangle=\sum_ic_i|\chi_i\rangle|\eta_i\ Rangle,\quad\langle\chi_i|\chi_j\rangle=\langle\eta_i|\eta_j\rangle=\delta_{ij}.\]
Obviously, this is singular value decomposition,\ (A=u^\dagger S v\), will \ (a_{jk}\) decomposition of the descendants back to the original, the number of non-0 \ (c_i\) is called the Schmitt-rank. The related concepts are also mixed-state purification, which will be seen as a subsystem for larger systems. The use of purification proves that the decomposition of the mixed state is not unique.

At this point, most of the quantum mechanics required for the computation has already been introduced, and some of the internal and EPR paradoxes in the mixed state are placed in the next chapter.

QI Chapter 1