### Foundations of Quantum Computing

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## 2.2. Foundations of Quantum Computing

The most widespread computational model in quantum computing – and arguably its simplest – is built on the qubit abstraction. As its name suggests, it is the quantum analogue of the classical bit i.e. a value that can take the values `0` or `1`.

Whilst we do not want to delve to deeply into the details of the physical realisations of qubits in real world architectures, it is important to note one fundamental difference with classical systems. Classical bit values (the famous `0`s and `1`s of our computers) are typically encoded using two voltages; another way of saying this is that bit values, and hence data, correspond to electrical currents on a chip. Gates, i.e. the lowest level of operations that can be applied to bits, then correspond to barriers that either let the electrical current flow through, or block it. Generalising slightly, we can hence picture a classical gate as a black box with `n` input wires going into the box and `m` output wires leaving it. For any combination of on and off voltages on the input wires, the box will turn on some of the output wires. The important points to take away from this classical state of affairs is that we can think of input and output bits (i.e. the input and output wires) as physically distinct objects (the wires).

This is not the case in implementations of qubits. Unlike their classical counterparts, quantum gates must be understood as operations that modify – or “mutate”, to borrow a term from programming languages – the physical qubits themselves. An input qubit to a gate is thus submitted to physical interactions that change its internal state. After the gate execution is completed, the qubits that held the input states now contain the output of the operation.

This has several profound implications for quantum computing. First and foremost, every quantum gate must have the same number of inputs as outputs. Most of the iconic classical gates (AND, OR, XOR etc.) are thus impossible to implement on a quantum computer without some adjustments1. This also means that the number of qubits must remain unchanged throughout the computation. A computation that starts with `n` qubits must also end in `n` qubits – and have `n` qubits at every point throughout the computation.

At this point, taking the preservation of qubits just described seriously, we should justifiably be asking how a quantum computation can even come to be at all, given that no qubit can be created out of thin air. In our attempt to remain blissfully ignorant of physical realities, we suggest to adopt the following abstracted mental model of qubits: qubits can neither be created nor deleted2, they simply i) exist at all times, and ii) can be reset to the 0 state.

For our convenience, we can then choose to ignore qubits that are of no importance to us. If all we need are `n` qubits, then will limit our considerations to these and pretend none other exist. Pushing further our myopic focus on qubits with a direct utility, we can also adjust the window of qubits of interest as we progress through the computation. If for instance a new qubit becomes useful halfway through our program execution, we can enlarge the set of qubits we are keeping track of and refer to this as “creating” a qubit. Conversely, it often happens that qubits become irrelevant, in which case we move them outside of our field of consideration and say that the qubits were discarded.

A final consequence of mutating qubits that we will highlight is that once a gate has been applied, the input states to the gate no longer exist! In other words, any state that we reach throughout our execution can only be used at most once. Here, your classical intuition might kick in:

> Let us just maintain a copy of the original state before modifying it!

This would indeed allow us to do more than one computation from a temporary value. No, big no no! This is a profound restriction (or property, depending on your point of view) with deep roots in the physics of quantum mechanics. This principle, known as the no-cloning theorem, is closely linked to many of the observations and properties that we have described so far.

### No-cloning theorem #

We are now entering the (short) section of the thesis where we discuss some ideas from quantum information theory – so depending on your disposition, either indulge yourself, or bear with me for a moment. The no-cloning principle Wootte., 1982W. K. Wootters and W. H. Zurek. 1982. A single quantum cannot be cloned. Nature 299, 5886 (October 1982, 802--803). doi: 10.1038/299802a0 provides a formal foundation for the vague statement “qubits live forever” that we made earlier. It is a fundamental tenant of quantum information, deserving of a more rigorous treatment that we are giving it here. We recommend that the curious reader refers themselves to more respectful references such as Nielsen, 2016Michael A. Nielsen and Isaac L. Chuang. 2016. Quantum Computation and Quantum Information (10th Anniversary edition). Cambridge University Press3.

No-cloning theorem: it is impossible to copy an arbitrary unknown state onto another (possibly known) qubit, or to copy a (possibly known) qubit to a qubit with unknown arbitrary state.

If we use ∣ψ⟩\ket{\psi}∣ψ⟩ to denote an arbitrary state and ∣0⟩\ket{0}∣0⟩ to denote a known state, the principle can be restated as: there are no quantum computations mapping ∣ψ⟩∣0⟩↦∣ψ⟩∣ψ⟩\ket{\psi}\ket{0} \mapsto \ket{\psi}\ket{\psi}∣ψ⟩∣0⟩↦∣ψ⟩∣ψ⟩, nor ∣ψ⟩∣0⟩↦∣0⟩∣0⟩\ket{\psi}\ket{0} \mapsto \ket{0}\ket{0}∣ψ⟩∣0⟩↦∣0⟩∣0⟩. The consequences of this are profound.

A consequence of the first half is what we alluded to in the previous section: any qubit states can only be used once in a computation. This statement is also what allows to justify why every quantum gate implementation, no matter the hardware specifics, will by necessity mutate its input qubits to produce the output states.

The second half of the statement is often also referred to as to “no delete” theorem. Indeed, if we view ∣ψ⟩\ket{\psi}∣ψ⟩ as a state encoding some data, we can interpret it as some amount of information. The state ∣0⟩\ket{0}∣0⟩, on the other hand, is a fixed state, and thus cannot store any amount of information. From the perspective of information theory, the map ∣ψ⟩∣0⟩↦∣0⟩∣0⟩\ket{\psi}\ket{0} \mapsto \ket{0}\ket{0}∣ψ⟩∣0⟩↦∣0⟩∣0⟩ thus destroys information: it turns an information storing left hand side into a product of ∣0⟩\ket{0}∣0⟩ states, devoid of any information.

We can also revisit the first map ∣ψ⟩∣0⟩↦∣ψ⟩∣ψ⟩\ket{\psi}\ket{0} \mapsto \ket{\psi}\ket{\psi}∣ψ⟩∣0⟩↦∣ψ⟩∣ψ⟩ and understand it from an information theoretic perspective as an attempt to create information out of thin air! Using this interpretation, the no-cloning theorem is thus the statement that quantum information is a preserved quantity in quantum computations: its amount will never increase, nor will it ever decrease.

### Reversibility #

That the amount of quantum information can never increase by transforming quantum states matches our intuition: if no new information is added from outside the system, then the total information that is encoded should not be increasing. Why however is it impossible to erase some information and thus reduce the total amount of it? The answer is reversibility: it is another foundation of quantum physics that every quantum of operation must be undoable, i.e. a computation must have an inverse operation that recovers the input when applied to the output.

If a quantum operation were thus to erase any information, then there would exist an inverse operation that creates information from nothing! The two halves of the no-cloning theorem as we presented it thus state the same principle once we consider that every operation must be reversible.

### Universality #

Finally, a third distinguishing property of quantum computation is how arbitrarily large computations can be generated from just single qubit gates and pairwise interactions between qubits (called entangling operations) Barenco, 1995Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin and Harald Weinfurter. 1995. Elementary gates for quantum computation. Physical Review A 52, 5 (November 1995, 3457--3467). doi: 10.1103/PhysRevA.52.3457. We call a set of gates that can be used to construct any arbitrary quantum computation a universal gate set.

This is a boon for hardware design, as manipulating single qubit systems is often much easier than controlling physical interactions between multiple entities. This decomposition into one and two-qubit gates means that the architecture i) does not need to support interactions between n>2 n > 2n>2 qubits, and ii) can be specialised and hand tuned to execute the two-qubit interaction of choice as faithfully as possible. Having a two-qubit gate as the entangling operation is not the only choice either. Some architectures such as neutral atoms choose instead to replace it with a global entangling operation that applies to many qubits simultaneously Evered, 2023Simon J. Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H. Li, Alexandra A. Geim, Tout T. Wang, Nishad Maskara, Harry Levine, Giulia Semeghini, Markus Greiner, Vladan Vuletić and Mikhail D. Lukin. 2023. High-fidelity parallel entangling gates on a neutral-atom quantum computer. Nature 622, 7982 (October 2023, 268--272). doi: 10.1038/s41586-023-06481-y, resulting in a universal gate set that is more convenient to implement experimentally in their system.

Gate set universality can be generalised further to approximate universality, which is at the centre of the development of error correcting codes. Indeed, it turns out that any quantum computations can be approximated to arbitrary precision using only discrete finite sets of one and two-qubit gates Kitaev, 2002Alexei Y. Kitaev, A. H. Shen and Mikhail N. Vyalyi. 2002. Classical and Quantum Computation. American Mathematical Society Dawson, 2006Christopher M. Dawson and Michael A. Nielsen. 2006. The Solovay-Kitaev algorithm. Quantum Information and Computation 6, 1 (January 2006, 81--95). doi: 10.26421/QIC6.1-6. This represents a significant simplification for error correction, as it removes the need for continuously parametrised gates and discretises the problem space.

We have just spent a long while discussing universality, reversibility and no-cloning. There is reason for it: these laws of physics that govern quantum computations and are absent from classical computer science turn out to be a great boon for quantum optimisation and compilation in general.

As we have just discussed, the existence of a wide variety of universal gate sets are degrees of freedom that the compiler can take advantage of. Transpilation from one universal gate set to another to enable programmers to target different hardware is one of the first and most fundamental functions of quantum compilers Sivara., 2020Seyon Sivarajah, Silas Dilkes, Alexander Cowtan, Will Simmons, Alec Edgington and Ross Duncan. 2020. t|ket⟩: a retargetable compiler for NISQ devices. Quantum Science and Technology 6, 1 (November 2020, 014003). doi: 10.1088/2058-9565/ab8e92.

Reversibility is also the source of a lot of flexibility when expressing quantum programs. Suppose the user wants to execute an operation A AA but it is more convenient, or the hardware is only capable of, executing a different gate B B B. Then using the inverse B−1 B^{-1}B−1 of B B B, it is always possible to rewrite the program as

where these diagrams should be read as operations to be executed from left to right. This is nothing but the mathematical trick of multiplying the left hand side with the identity operation expressed as B−1⋅B B^{-1} \cdot B B−1⋅B.

Now of course this rewrite is only sensible if the operation A∘B−1 A \circ B^{-1}A∘B−1 is reasonably cheap to perform4. There are plenty of instances where this is indeed the case. Morally, the quantum compiler always has the freedom to execute any quantum operation – at the risk of course of producing very inefficient code – given that reversibility always guarantees that the operation can be reversed and the competition undone whenever necessary.

No-cloning is also a very useful guarantee that the compiler can make use of. In chapter X we will see that it greatly simplifies pattern matching, useful to quickly identify all possible optimisations. More generally, no-cloning restricts the set of programs that the compiler must consider, resulting in simply compiler logic and better performance. we will see examples of this in chapter 3.

### The quantum circuit representation #

We could not conclude our overview of the basics of quantum computing without a mention of the quantum circuit, a representation of quantum computation that is ubiquitous in the field. With the understanding that we have gained in the course of this section, the two building blocks of the circuit model and the conventions around their graphical representation should be of no surprise to the reader:

1. Qubits are represented by straight, horizontal lines. Their evolution through time can be followed along the line from left to right: At the leftmost point on the line, the qubit is in its input state; when the qubit reaches its rightmost point, operations have mutated it into the output state of the circuit.
2. Gates on qubits are boxes placed vertically across one or multiple qubit lines. The qubits it is on represents the set that the gate may act on (and mutate), whereas the left-to-right ordering of the gates reflects their ordering in time.

A simple circuit composed of two qubits and three gates G GG, H H H and K K K could for instance look like this:

In this case, G GG would be executed before H H H and K K K; G GG would act on both qubits whereas H H H and K K K would only modify the first and second qubit respectively. Note that there is no ordering specified between H H H and K K K: because they act on disjoint sets of qubits, their relative ordering makes no difference. It is thus common to display them as acting at the same time. We could have equivalently chosen to draw them as:

All these circuits represent the same computation.

Certain quantum gates are particularly useful and appear very regularly in practice. These have standard names that are widely used in the field. The most common single qubit gates are arguably the Hadamard, represented in circuits by a H H H box, and the X X X, Y Y Y and Z Z Z-axis rotations, drawn as R x(θ)R_x(\theta)R x​(θ), R y(θ)R_y(\theta)R y​(θ) and R z(θ)R_z(\theta)R z​(θ) boxes respectively. Note that rotation gates are parametrised by an angle θ∈[0, 4 π)\theta \in [0, 4\pi)θ∈[0, 4 π) that must be specified to actually execute the rotation.

There are also commonly used multi-qubit gates. For these, it becomes slightly awkward to draw them as boxes, as they may act on qubits that are not drawn next to eachother in the circuit5. As a solution to this, common gates were given representations that do not spell out their name, but clearly mark which qubit they are action on, and in what order. Here are the representations of three of the most famous ones, in order: the CX (also known as CNOT) gate, the CZ and the Toffoli:

{ "qubits": [{ "id": 0 }, { "id": 1 }, { "id": 2 }], "operations": [ { "gate": "X", "isControlled": true, "controls": [{ "qId": 1 }], "targets": [{ "qId": 2 }] }, { "gate": "Z", "isControlled": true, "controls": [{ "qId": 0 }], "targets": [{ "qId": 1 }] }, { "gate": "X", "isControlled": true, "controls": [{ "qId": 0 }, { "qId": 1 }], "targets": [{ "qId": 2 }] } ] }

You will probably notice that there seems that there seems to be a system to this graphical notation. There is, but unfortunately explaining it would require us to discuss Pauli matrices, commutation relations, and quickly lead us astray. The references at the end of this chapter6 are a good starting point for further reading.

1. The NOT gate is the notable exception to this. It is often found in quantum programs and called X. ↩︎

2. This is indeed true physically: the carriers of quantum information, typically atoms or photons, live forever in the absence of interactions with their environment. However we would be seriously deluding ourselves if we believed that the control systems we use to manipulate and keep these particles trapped can do so for any significant amount of time. Instead, experimentalists have to constantly come up with creative ways to stop the qubits from escaping or interacting with their surroundings (and destroying themselves in the process). ↩︎

3. or, if you really must: Coecke, 2017Bob Coecke and Aleks Kissinger. 2017. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press. doi: 10.1017/9781316219317. ↩︎

4. The ∘\circ∘ is maths for denoting the composition of functions, so unlike the left-to-right diagram, it must be read from right to left. Is your head spinning yet? ↩︎

5. This becomes immediately apparent if you attempt to draw a a gate that should act on the first and third qubit line of a circuit, but leave the second one untouched. ↩︎

6. Or my email! ↩︎

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