# What is Quantum Computing?

Ian Hellström | 29 August 2022 | 9 min read

Whereas current computers use bits that are each either zero or one, quantum bits can be both zero and one. This quirky property of quantum physics can be used to our advantage in quantum computers.

Quantum bits (a.k.a. qubits) are typically in superpositions, or linear combinations, of quantum states that represent such 0s and 1s. We choose the assignment of quantum states to qubit states |0⟩ and |1⟩, although we typically pick the system’s ground state to represent |0⟩ and the first excited state as |1⟩. But that is merely a matter of definition.

This difference between the classical and quantum world can be visualized as follows. Classically, we have two mutually exclusive options for bits: 0 or 1. These can be pictured as two points with no connecting line in-between. Qubits states |ψ⟩ can lie anywhere on a sphere (a.k.a. Bloch sphere).

Why are qubits constrained to a sphere?

An arbitrary superposition of |0> and |1> can be written as |ψ> = α|0> + β|1>, in which α and β are complex numbers. Physically, the squared magnitudes of these so-called amplitudes α and β represent the probabilities of observing either |0> or |1> upon measurement. Hence, |α|2 + |β|2 = 1 to ensure all probabilities add up to 1. Each state |ψ> is therefore represented by two complex numberers (α, β), which means we have four (real) degrees of freedom per qubit. The normalization constraint reduces that to three. Qubits are therefore stuck on a three-dimensional sphere.

Another way to think about superposition is to view zeros and ones as heads or tails of a coin. Classically, the state of the coin is either heads or tails. Simple.

Quantum mechanically, the state of a single qubit is a superposition of heads and tails, as if the coin were in a tossed state. While tossed, it is neither heads or tails but rather both at the same time. At least until we ‘measure’ the coin, at which point the superposition is destroyed, and we measure heads or tails with probability 0.5 each.

That superposition is not due to our inability to know better or because of hitherto undiscovered principles of nature that behave more classically. In fact, Bell’s theorem and subsequent experimental verifications have once and for all disproved the existence of such classically hidden variables:

The universe around you is inescapably probabilistic. It evolves in a deterministic fashion through Schrödinger evolution. But when we measure things, we measure results with probabilities. And those probabilities cannot be explained through some underlying classical dynamics. […] [T]his property of probabilistic evolution, or probabilistic measurement, is an inescapable and empirically verified property of the reality around us. Allan Adams (2013)

In other words, it is fundamentally impossible to peek at the coin’s sides as it spins in the air or look inside the box of Schrödinger’s infamous cat – without ruining the delicate quantum state. God may not play dice, but the universe definitely is a quantum casino.

## What makes quantum computers special?

The key to the unlocking the value of quantum computers is entanglement. Entangled qubits are inseparable superpositions of individual qubits; these multi-qubit states cannot be decomposed into a (tensor) product of individual qubit states. Phrased differently, we can encode correlations among entangled qubits, process these qubits individually, and extract the desired information, often in unique and more efficient ways than is possible classically, that is, with regular bits.

This entanglement lies at the heart of the infamous ‘spooky action at a distance’ in the EPR paradox, and it too is fundamental to quantum mechanics.

### Quantum parallelism

Each execution of an algorithm on a quantum computer gives 1 out of 2n possible outcomes for n qubits. We can apply gates to all n qubits at the same time and affect all 2n possible states, which means computations can be done in parallel. This parallelism is free, courtesy of quantum physics. However, due to the intrinsically probabilistic nature of quantum mechanics and the destructive properties of measurements, quantum computers require multiple runs.

Because a measurement is required to extract information, qubits are single use. They cannot be cloned either, although their states can be prepared repeatedly, which is why they can be used as samplers. Hence, quantum computers will not replace classical computers in the future, but rather become co-processors, not unlike GPUs and TPUs. Such accelerators are known as quantum processing units or QPUs. QPU speed-ups of certain algorithms are not due to faster, specialized hardware, as in the case of GPUs and TPUs, but due fundamental improvements in computational complexity, courtesy of quantum physics.

Famous quantum algorithms include Grover's search algorithm, which requires O(√n) instead of O(n) evaluations, and Shor's factorization algorithm, which runs in polynomial time. The quantum Fourier transform, which lies at the heart of Shor's algorithm, requires O(n2) quantum gates for n qubits instead of O(n 2n). This represents an exponential speed-up! Similar impressive speed-ups exist for a large collection of quantum algorithms.

As samplers, quantum accelerators generate candidate solutions, such that good candidates have a higher probability of being measured.

### Feature space

Another crucial feature of quantum computers is that superposition enables a larger feature space to be explored, which is especially of interest to machine learning applications and optimization problems, for which it can be advantageous to fit a richer set of functions to the data. A fault-tolerant 30-qubit computer, which is well within reach within a decade or so, can explore a space of 230, or more than a billion, features. Each additional qubit doubles that figure. Such exponential growth is hard to beat, even with distributed computations on state-of-the-art GPUs.

Why qubits and not qudits?

While there are potential benefits to quantum digits or qudits, higher excitations typically decay more rapidly. Maintaining such quantum states is therefore more involved, although metastable excited states do exist.

Trapped ions have been shown to support qudits. Still, the integration of such QPUs with existing CPUs requires new hardware beyond merely a quantum chip, which may ultimately prove prohibitively expensive and complex.

## What can quantum computers be used for?

As black-box accelerators to classical algorithms, quantum computers promise to provide significant speed-ups for many different problems, such as logistics optimization (Airbus), multi-flow classification (Aker BP), computational fluid dynamics (BAE Systems), weather modelling (BASF), credit valuations, (BBVA), vehicle sensor placement (BMW), portfolio management (CaixaBank), and derivative pricing (Goldman Sachs).

Furthermore, we can view quantum computers as the API to the quantum world:

[N]ature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical[.]Richard Feynman (1982)

This approach leads to promising avenues of research and development in quantum physics and quantum chemistry simulations, quantum metrology, and even programmable quantum experiments.

Examples of such use cases include material discovery (Boeing), catalyst design (ExxonMobil), battery development (Daimler), cancer treatment biomarker discovery (CrownBio, JSR Life Sciences), carbon capture and storage (Total), drug design and discovery (Boehringer-Ingelheim, GSK, Roche).

The fact that qubits cannot be cloned implies that secure quantum communication channels can be devised. Quantum communication protocols exploit entanglement to ensure no unauthorized third party can listen in and go undetected. The presence of eavesdroppers is measurable to both the sender and the recipient. It is thus impossible for such intruders to 'tap a wire' and remain unnoticed.

The fundamental impossibility of cloning is a direct consequence of linearity of Schrödinger's equation and unitarity of its time evolution operator. Linearity implies that superpositions of solutions to Schrödinger's equation are also valid solutions. Unitarity implies is that probabilities are preserved.

## What are quantum computers made of?

There are currently 7 paths towards quantum computers:

1. Superconductors
2. Ion traps
3. Photonics
4. Topological superconductors
5. Spin qubits in semiconductors (a.k.a. quantum dots)
6. Neutral atoms
7. Quantum annealers

Quantum computers made up of even more exotic qubits, such as chemical qubits (e.g. quantum functional groups), are being researched. While these may exist in the lab, no commercial solutions of such exotic quantum computers exist yet.

### Superconductors

Superconductors are by far the most common and mature approach, pursued by the likes of Alice & Bob, Bleximo, Google, IBM, Intel, IQM, OQC, Rigetti, and many more.

These quantum computers require temperatures near absolute zero, which makes scaling systems up tricky as it requires bulky dilution refrigerators. There are three basic superconducting qubit types: charge, flux, and phase. In so-called transmon (charge) qubits, the lowest two states are mapped to the qubit states |0⟩ and |1⟩. The decoherence time of qubits is pretty short, which means it is currently hard to create complex algorithms with many layers of quantum gates. Quantum error correction schemes for such quantum computers require many physical qubits for each logical one.

### Trapped ions

AQT, Honeywell (Quantinuum), IonQ, and Universal Quantum are known for quantum computers based on ion traps. Infineon have recently entered the fray for trapped-ion quantum computers with integrated optics, too. The company also has a partnership with Oxford Ionics to develop QPUs.

Charged particles (ions) are trapped inside a time-dependent electromagnetic field, arranged on a chain, and cooled down, so the ions act and oscillate as a collective (phonons). Lasers control both the excitations of the ion chain and the motion of the chain itself. The lowest two phonon states correspond to the qubit states.

### Photonics

Photonic quantum computers rely on photons or light, which has the advantage that if the technology can be made to work reliably, it can be scaled up relatively easily at room temperature. Qubit states correspond to the modes of the light, such as polarization, frequency, or any other degree of freedom. Instead of qubits, photonic quantum computers are typically measured in qumodes, for which there is no one-to-one translation.

ORCA Computing, PsiQuantum, QCI, Quandela, QuiX Quantum, and Xanadu play in this arena.

### Topological superconductors: Majorana bound states

Microsoft work on topological quantum computers. The qubits are Majorana bound states in a semiconductor-superconductor junction. Particle-hole symmetry protects these bound states from disturbances. Unfortunately, these machines also require massive cryogenic equipment, just like superconducting and trapped-ion quantum computers.

Note such Majorana quasiparticles have yet to be demonstrated conclusively. This technology is therefore very promising but highly speculative.

### Quantum dots: spin qubits in semiconductors

Intel are also pursuing quantum dots, because the fabrication process is similar to those of semiconductors. Another spin qubit player is Diraq.

Spin states of electrons in a quantum dot encode the qubits. In that sense, these qubits are truly a two-level system, and not a multi-level system constrained to the two lowest energy levels.

### Spin qubits in NV centres

Spin qubits can also be created in nitrogen-vacancy (NV) centres in diamond. Such quantum computers require temperatures on the order of only a few Kelvin. Spins of electrons trapped at vacancies are used as qubits, although the coupling to nuclear spins gives rise to more qubits. Electron states are controlled with microwave pulses, whereas nuclear spins can be controlled through radio frequencies. Quantum Brilliance are working on commercial quantum computers based on NV centres.

### Neutral atoms

Quantum computers based on cold neutral atoms consist of magneto-optically trapped atoms, for which the energy levels or nuclear spins map to qubits. These atoms can be manipulated into complex two- and three-dimensional arrangements, although they require extremely low temperatures and vacuums.

Such quantum computers are the purview of Atom Computing, ColdQuanta, Pasqal, planqc, and QuEra Computing.

### Quantum annealers

D-Wave are known for their quantum annealers. While these machines are made from superconductors, they are fundamentally different from superconducting quantum computers in that they require an equal superposition of states as initialization, after which the system evolves adiabatically. Such quantum annealers are especially suited for optimization problems.

Note that D-Wave are currently also working on a universal gate-based superconducting quantum computer.

Are quantum computers digital?

Adiabatic quantum computers and quantum annealers are analogue, as they evolve the quantum state continuously. Gate-based quantum computers are digital, as they transform quantum states in a series of discrete operations. Qubits on such gate-based chips are manipulated with lasers, microwave and radio frequency pulses, and electromagnetic fields. These are all analogue signals. To send signals to the quantum chip, a DAC (digital-to-analogue converter) is required to transform instructions from the host CPU to the QPU. Likewise, an ADC (analogue-to-digital converter) is needed for storing the data after reading out the quantum register and subsequent further processing in the CPU.

## Outlook

Quantum computers promise to revolutionize computing in the twenty-first century. If you already want to try out what can be done with quantum computers and simulators today, check out Zoose Quantum, a batteries-included Docker image for quantum computing in JupyterLab. It comes with various extensions to make the experience smoother. It requires no setup of virtual environments or the like. Just Docker.

In a future post, I shall review various online learning opportunities for quantum computing in depth. That way, you can make an informed decision on the best quantum eduction for you.