## 1.10 *Remarks and exercises*

### 1.10.1 A historical remark

Back in 1926, Max Born simply postulated the connection between amplitudes and probabilities, but did not get it quite right on his first attempt.
In the original paper^{20} proposing the probability interpretation of the state vector (wavefunction) he wrote:

… If one translates this result into terms of particles only one interpretation is possible.

\Theta_{\eta,\tau,m}(\alpha,\beta,\gamma) [the wavefunction for the particular problem he is considering] gives the probability^* for the electron arriving from thez direction to be thrown out into the direction designated by the angles\alpha,\beta,\gamma …

^* Addition in proof: More careful considerations show that the probability is proportional to the square of the quantity\Theta_{\eta,\tau,m}(\alpha,\beta,\gamma) .

### 1.10.2 Modifying the Born rule

Suppose that we modified the Born rule, so that probabilities were given by the absolute values of amplitudes *raised to power p* (for some

Recall that the *only* isometries, except for *one* special case!
For *see* this result.

In particular, the image of the unit sphere must be preserved under probability preserving operations.
As we can see in Figure 1.7, the

### 1.10.3 Complex numbers

Complex numbers have many applications in physics, however, not until the advent of quantum theory was their ubiquitous and fundamental role in the description of the actual physical world so evident. Even today, their profound link with probabilities appears to be a rather mysterious connection. Mathematically speaking, the set of complex numbers is a field. This is an important algebraic structure used in almost all branches of mathematics. You do not have to know much about algebraic fields to follow these lectures, but still, you should know the basics. Look them up.

- The set of rational numbers and the set of real numbers are both fields, but the set of integers is not. Why?
- What does it mean to say that the field of complex numbers is
**algebraically closed**? - Evaluate each of the following quantities:
1+e^{-i\pi}, \quad |1+i|, \quad (1+i)^{42}, \quad \sqrt{i}, \quad 2^i, \quad i^i. - Here is a simple proof that
+1=-1 :1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i^2=-1. What is wrong with it?

### 1.10.4 Many computational paths

A quantum computer starts calculations in some initial state, then follows ^{21}

### 1.10.5 Distant photon emitters

Imagine two distant stars, A and B, that emit *identical* photons.
If you point a single detector towards them you will register a click every now and then, but you never know which star the photon came from.
Now prepare two detectors and point them towards the stars.
Assume the photons arrive with the probability amplitudes specified in Figure 1.8.
Every now and then you will register a coincidence: the two detectors will fire.

- Calculate the probability of a coincidence.
- Now, assume that
z\approx \frac{1}{r}e^{i\frac{2r\pi}{\lambda}} , wherer is the distance between detectors and the stars. How can we use this to measurer ?

### 1.10.6 Physics against logic?

Now that we have poked our heads into the quantum world, let us see how quantum interference challenges conventional logic and leads to qualitatively different computations.
Consider the following task (which we will return to a few more times in later chapters): design a logic gate that operates on a single bit such that, when it is followed by another, identical, logic gate, the output is *always* the negation of the input.
Let us call this logic gate **the square root of \texttt{NOT}**, or

Figure 1.9 shows a simple computation, two identical computational steps performed on two states labelled as

Write a

### 1.10.7 Quantum bomb tester

You have been drafted by the government to help in the demining effort in a former war-zone.^{22}
In particular, retreating forces have left very sensitive bombs in some of the sealed rooms.
The bombs are configured such that if even one photon of light is absorbed by the fuse (i.e. if someone looks into the room), the bomb will go off.
Each room has an input and output port which can be hooked up to external devices.
An empty room will let light go from the input to the output ports unaffected, whilst a room with a bomb will explode if light is shone into the input port and the bomb absorbs even just one photon — see Figure 1.10.

Your task is to find a way of determining whether a room has a bomb in it without blowing it up, so that specialised (limited and expensive) equipment can be devoted to defusing that particular room. You would like to know with certainty whether a particular room had a bomb in it.

- To start with, consider the setup in Figure 1.11, where the input and output ports are hooked up in the lower arm of a Mach–Zehnder interferometer.
^{23}- Assume an empty room.
Send a photon to input port
|0\rangle . Which detector, at the output port, will register the photon? - Now assume that the room does contain a bomb.
Again, send a photon to input port
|0\rangle . Which detector will register the photon and with which probability? - Design a scheme that allows you — at least some of the time — to decide whether a room has a bomb in it without blowing it up. If you iterate the procedure, what is its overall success rate for the detection of a bomb without blowing it up?

- Assume an empty room.
Send a photon to input port

Assume that the two beam splitters in the interferometer are different. Say the first beam-splitter reflects incoming light with probability

r and transmits with probabilityt=1-r , and the second one transmits with probabilityr and reflects with probabilityt . Would the new setup improve the overall success rate of the detection of a bomb without blowing it up?There exists a scheme, involving many beam-splitters and something called the

**quantum Zeno effect**, such that the success rate for detecting a bomb without blowing it up approaches 100%. Try to work it out, or find a solution on the internet.

### 1.10.8 More time, more memory

A quantum machine has

Suppose you are using your laptop to add together amplitudes pertaining to each of the paths.
As

### 1.10.9 Quantum Turing machines

The classical theory of computation is essentially the theory of the universal Turing machine — the most popular mathematical model of classical computation. Its significance relies on the fact that, given a large but finite amount of time, the universal Turing machine is capable of any computation that can be done by any modern classical digital computer, no matter how powerful. The concept of Turing machines may be modified to incorporate quantum computation, but we will not follow this path. It is much easier to explain the essence of quantum computation talking about quantum logic gates and quantum Boolean networks or circuits. The two approaches are computationally equivalent, even though certain theoretical concepts, e.g. in computational complexity, are easier to formulate precisely using the Turing machine model. The main advantage of quantum circuits is that they relate far more directly to proposed experimental realisations of quantum computation.

### 1.10.10 Polynomial = good; exponential = bad

In computational complexity the basic distinction is between polynomial and exponential algorithms.
Polynomial growth is good and exponential growth is bad, especially if you have to pay for it.
There is an old story about the legendary inventor of chess who asked the Persian king to be paid only by a grain of cereal, doubled on each of the 64 squares of a chess board.
The king placed one grain of rice on the first square, two on the second, four on the third, and he was supposed to keep on doubling until the board was full.
The last square would then have ^{24}
The moral of the story: if whatever you do requires an exponential use of resources, you are in trouble.

### 1.10.11 Big O

In order to make qualitative distinctions between how different functions grow we will often use the asymptotic big-^{25} which means that

When we say that

f(n)=O(\log n) , why don’t we have to specify the base of the logarithm?Let

f(n)=5n^3+1000n+50 . Isf(n)=O(n^3) , orO(n^4) , or both?Which of the following statements are true?

n^k=O(2^n) for any constantk n!=O(n^n) - if
f_1=O(g) andf_2=O(g) , thenf_1+f_2=O(g) .

### 1.10.12 Imperfect prime tester

There exists a randomised algorithm which tests whether a given number ^{26}
The algorithm always returns

### 1.10.13 Imperfect decision maker

Suppose a randomised algorithm solves a decision problem, returning ^{27}.
If we are willing to accept a probability of error no larger than

- If we perform this computation
r times, how many possible sequences of outcomes are there? - Give a bound on the probability of any particular sequence with
w wrong answers. - If we look at the set of
r outcomes, we will determine the final outcome by performing a majority vote. This can only go wrong ifw>r/2 . Give an upper bound on the probability of any single sequence that would lead us to the wrong conclusion. - Using the bound
1-x\leqslant e^{-x} , conclude that the probability of our coming to the wrong conclusion is upper bounded bye^{-2r\delta^2} .

Max Born, “Zur Quantenmechanik der Stoßvorgänge”,

*Zeitschrift für Physik***37**(1926), 893–867.↩︎1+z+z^2+\ldots + z^n= \frac{1-z^{n+1}}{1-z} ↩︎This is a slightly modified version of a bomb testing problem described by Avshalom Elitzur and Lev Vaidman in

*Quantum-mechanical interaction-free measurement*, Found. Phys.**47**(1993), 987-997.↩︎One light year (the distance that light travels through a vacuum in one year) is

9.4607\times10^{15} metres.↩︎f=O(g) is pronounced as “f is big-oh ofg ”.↩︎Primality used to be given as the classic example of a problem in

\texttt{BPP} but not\texttt{P} . However, in 2002 a deterministic polynomial time test for primality was proposed by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. Thus, since 2002, primality has been in\texttt{P} .↩︎This result is known as the

**Chernoff bound**.↩︎