Part III: Universal quantum computing

Preface

Quantum Computer is such a computing device that can efficiently simulate any other physical system

In this part,we'll discuss what a quantum computer is,why they're useful,and whether they can be used to efficently simulate any other physical system

The quantum computing model

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics. — David Deutsch

In a general quantum computation, you start out with many qubits，In a general quantum computation, you start out with many qubits.
You apply quantum gates of various kinds, in particular, single-qubit gates and CNOT gates.
And at the end of the circuit you read out the result by measuring in the computational basis
for example:

We can summarize the three steps in a quantum computation as follows:

1. Start in a computational basis state.
2. Apply a sequence of CNOT and single-qubit gates.
3. To obtain the result, measure in the computational basis. The probability of any result, say 00…0, is just the square of the absolute value of the corresponding amplitude.

That’s all a general quantum computation is!

From the perspective of programmer

you can think of this as like the way programming language designers introduce higher-level abstractions to help people design different kinds of programs

In principle,those abstractions can always be reduced down to the level of AND and NOT gates,which is the basis of other ideads

In fact,we don't need to master all the higher-level abstractions
Reason:

• humanity doesn't yet know what the higher-level abstractions are
• the list of higher-level abstractions is inexhaustible(we will always discover new abstractions forever)

Different models

As a matter of fact,there are many different models of quantum computation,like, qutrit,which has three computational basis states.\(\vert 0 \rangle,\vert 1 \rangle,\vert 2 \rangle\)

Nonetheless,different models are all mathematically equivalent to one another,including quantum circuit model.

The diverse models stimulate different ways of thinking and give rise to different ideas——————It's the value that they all exist

multiples of the identity

\[\begin{bmatrix} e^{i\theta} & 0\\ 0 & e^{i\theta} \end{bmatrix}=e^{i\theta} \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}=e^{i\theta}I \]

When θ is a real number this is a unitary matrix, and so a valid quantum gate.The effect of the gate is simply to multiply the state of the quantum computer by \(e^{i\theta}\),which is called a global phase factor

But you find that they have no impact on the results of the computation,
because:

• the identity does effect the vector.
• doesn't change the squared amplitudes of the computational basis states
• has no impact on the measurement probabilities at the end of the computation.

so people often choose not to think of them

What are quantum computers good for?

Quantum Computers expand the range of operations available when computing.

To achieve it,the quantum computers need to be at leasted as capable as classical computers.
Fortunately,it's possible to convert any classical circuit into a quantum circuit
in terms of some standard universal set,such as the AND and NOT gates

we know the NOT gates can be turned into an X gate,But to be frank,we can't find the AND gate---turn\(\vert x,y\rangle\) into
a single qubit\(\vert x\wedge y \rangle\)

• the reason: Not only is it not unitary, it’s not even close: a unitary gate with two qubits as input necessarily has two qubits as output.

the Toffoli gate

However,the Toffoli gate,a three-qubit quantum gate,can serve as the AND gate.

If the target starts out as z=0
z=0, then you can see that the target output is just \(x\wedge y\),conversely,the target output is just \(\neg x \wedge y\)--it is the NAND,the NOT of \(x \wedge y\)

A wrinkle in all this is that the Toffoli gate isn’t in our standard set of basic quantum gates. However, it’s possible to build the Toffoli gate up out of CNOT and single-qubit unitary gates. One way of doing the breakdown is shown below:

people simply trying lots and lots of different ways of implementing the Toffoli gate,just done by pure brute force.

Show that the inverse of the Toffoli gate is just the Toffoli gate:

The Toffoli gate (CCNOT) is its own inverse. Here's the proof:

Toffoli Gate Definition:

• Acts on 3 qubits (2 controls, 1 target).
• Flips the target qubit iff both controls are (|1\rangle).
• Matrix representation:

\[ T = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix} \]

Self-Inverse Property:

Applying the Toffoli gate twice undoes its effect:

\[T \cdot T = T^2 = I \quad (\text{Identity matrix}). \]

This is because flipping a qubit twice returns it to its original state.

General Case:

For arbitrary inputs \(|a, b, c\rangle\):

\[T(T(|a, b, c\rangle)) = T(|a, b, c \oplus (a \land b)\rangle) = |a, b, c \oplus (a \land b) \oplus (a \land b)\rangle = |a, b, c\rangle. \]

No, really, what are quantum computers good for?

It’s comforting that we can always simulate a classical circuit – it means quantum computers aren’t slower than classical computers – but doesn’t answer the question of the last section: what problems are quantum computers good for?

Can we find shortcuts that make them systematically faster than classical computers?

It turns out there’s no general way known to do that. But there are some interesting classes of computation where quantum computers outperform classical.

quantum computers will be simulating other quantum systems，unfortunately, classical computers are terrible at simulating quantum systems.
Given the situation，you will understand the reason

• For a molecule with n atoms, treating each atom as a qubit (though not accurate) would necessitate \(2^n\) amplitudes. In reality, atoms are more complex, requiring \(k^n\) amplitudes where \(k≥2\), and k can be in the hundreds for generic simulations. Even for simple molecules, this results in trillions of amplitudes, increasing rapidly with each additional atom. For example, k=100 and n=10 would require 100 million trillion amplitudes. Storing and updating these amplitudes is computationally intensive, making classical simulation impractical.

So with the quantum computation,many of these problems will become vastly easier

And the quantum computer will need just a small number of extra qubits compare to the classical computer
One way of thinking of this is as a loose quantum corollary to Moore’s law:

The quantum corollary to Moore’s law: Assuming both quantum and classical computers double in capacity every few years, the size of the quantum system we can simulate scales linearly with time on the best available classical computers, and exponentially with time on the best available quantum computers.

In the long run, quantum computers will win, and win easily.

Really answer the question

That answer is to list various algorithmic problems that we have some evidence can be solved faster on a quantum computer than on a classical computer.
The most famous example is Peter Shor’s beautiful quantum factoring algorithm

One exception to humanity’s private thoughts lack of public reflection is a brief discussion in Ronald de Wolf’s thoughtful essay The Potential Impact of Quantum Computers on Society

Are quantum computers really universal devices?

The eternal mystery of the world is its comprehensibility… The fact that it is comprehensible is a miracle. — Albert Einstein

This is an open problem.Nobady knows the answer.
Part of the trouble in answering the question is that humanity hasn’t yet discovered the final fundamental laws of physics

Can we use quantum computers to efficiently simulate general relativity and the standard model?

The standard model is an example of a particular type of quantum mechanical theory called a quantum field theory

John Preskill and his collaborators have written a series of papersFor a review of progress see: John Preskill, Simulating quantum field theory with a quantum computer (2018). explaining how to use quantum computers to efficiently simulate quantum field theories. Those papers don’t yet simulate the full standard model, but they do make considerable progress. It remains an exciting open problem, albeit a problem where much encouraging progress has been made.

significant issue

A quantum computer isn't a single device but can have many quantum circuits.

However, the concept of a universal quantum Turing machine addresses this by being a single device capable of simulating any quantum circuit.

As the quantum circuit model is more commonly used, interested readers are directed to a paper by Bernstein and Vazirani for further information

The Universality of Computation and the Laws of Physics

The existence of universal computers might seem taken for granted, but there's no inherent logical reason why a single machine should be able to efficiently simulate every other physical system.

there is a sense in which matter has universal properties: if protons, neutrons, and electrons could be rearranged arbitrarily, a car could indeed be transformed into a surfboard, trolley, or a small part of a rainforest.

This highlights the intriguing universality of matter.
The universe appears to allow for such universal machines, which is a remarkable fact.

Exploring sets of physical laws where such universal machines are impossible could provide deeper insights into our own universe, even though it might seem pointless to imagine other possible universes. Such thought experiments often lead to a better understanding of our reality.

There is a strange loop here: the laws of physics determine what computations can be performed, and yet the computations that can be performed are powerful enough to describe the laws of physics.

This description can then be used to efficiently simulate any physical system. The top part of this loop is almost tautological, but the bottom half is extraordinary.

There is no a priori reason why the laws of physics should enable the existence of machines capable of simulating physical systems.

While anthropic arguments might suggest that since we exist and are doing physics successfully, such machines must exist, this is not a satisfactory explanation. The fact that the world is comprehensible at all remains a miracle, as Einstein noted.

Final Reflections

We’ve worked through all the basics of the quantum computing model, but we haven’t yet used it in a full-on application – the sort of application which make people excited about quantum computing. But there will soon be available two considerably shorter(!) followup essays, explaining the quantum search algorithm and quantum teleportation. Someone who understands all three essays will have a good understanding of elementary quantum computing