Quantum Computing
Quantum Computers are commonly understood as gate based processors which can perform logical operations on
qubits. The hope is that once successfully engineered they will be able to execute specific tasks much faster and more accurately
than classical, transistor-based computers. Examples of such tasks, Grover's and Shor's algorithms, are elaborated on in sections 3 and 4.
Additionally, they can solve physical problems- a discussion and example of which is laid out in section 6.
Progress in manufacture and implementation is ongoing, with the promised 'Quantum Advantage' seemingly ever-closer yet always behind a slew of obstacles. A significant hinderance is quantum decoherence, wherein the
computer's quantum state couples with its environment and loses information. In more complex terms - the quantum gates become irreversible. To handle this, quantum error correction codes are implemented. A simplified version is laid out in section 5.
There are many different types of qubit - such as ionic, neutral atom, photonic and superconducting- each with their advantages and limitations. The physics behind superconducting qubits
is described in section 2, and an interesting application of this type of qubit is described imminently.
Quantum Annealing
Introduction
Quantum annealers are machines which use qubits, but not gates, to perform optimisation. The versatility of (error free) quantum computers is much more than that of a Quantum annealer, but still annealers promise to perform practical tasks at a faster rate than classical computers, and in fact already have (see the end of this section). Moreover, Quantum annealing is a heuristic, meaning that it generally offers a 'good enough' solution, not the best one - yet this is not so much of an obstacle and more provides a sort of wiggle room not applicable to the gate-based QPUs. The physics behind annealers and their applications are more concisely laid out in the ensuing subsections.
General Theory
Annealers are suited to solving Quadratic Unconstrained Binary Optimisation (QUBO) problems, which are combinatorial optimisation problems with an objective function of binary variables \(x_i \in {0,1}\) with only linear and quadratic terms. The objective function is optimised by the minimising combination of the set of binary variables. The objective function takes the form: \[ \operatorname{Obj}(\mathbf{x})=\sum_{i,j=1}^N Q_{i, j} x_i x_j+\sum_{i=1}^N c_i x_i \] where the first sum on the right hand side encapsualtes the quadratic terms and the latter the linear terms.
The first limitation of this condition is that practical problems tend to have contstraints, ie conditions that the variables must adhere to. A simple example of such a problem is the travelling salesperson problem, wherein one wants to find the shortest route a saleperson can take which visits all of the locations in a town, or equivalently nodes on a graph, at least once. The contraint here is that the salesperson must only be in a single location at each stage of the trip. For example with three locations (ie variables) a,b,c one requires \(a+b+c = 1\) where \(a=1\) is taken to be the truth condition. This is a linear constraint and hence can be added as a penalty term to the objective function: \[E_k = (a+b+c -1)^2 = P_k^2\] since this term will be minimised when only one of a,b,c are equal to 1. With multiple linear constraints \(P_k(\mathbf{x})\), the objective function undergoes the change: \[ \operatorname{Obj}(\mathbf{x}) \mapsto \operatorname{Obj}(\mathbf{x})+\sum_{k}\lambda_k P_k(\mathbf{x})^2 \] where the \( \lambda_k\) is tuned so that the \(P_k\) are dominant and combinations which optimise \( \operatorname{Obj}(\mathbf{x}) \) will not violate the constraints. A more detailed discussion of problem formulation and constraints can be found in this review paper and on D-waves website.
The problem here is of course not all constraints may be linear, and many problems go beyond quadratic order. Hence the annealer is limited to only a set of specific problems. The objective function is equivalent a graph \(G(V,E)\) with edges \(E\) and vertices \(V\), where the edges have weights \(Q_{i,j}\), and the vertices correspond to the variables \(x_i\) with biases \( c_i\). This graph is embedded into the physical arrangement of qubits in the annealer, which themselves also correspond to a graph, as shown in Figure. 1.

When one thinks of the objective function as an energy, the physical parallels become clear. Systems naturally want to minimise their energy - a ball will roll down a hill to minimise its graviational potetntial energy, and hot tea goes cold. Thermal annealing, a 6000 year old process, relies on this principle to remove defects in metals. Simulated classical annealing is a widely used heuristic algorithm which simulates thermal effects in order to minimise an objective function in a similar way. This process relies on virtually heating the system then slowly reducing the temperature so the variables settle into values or 'states' which minimise the energy, relying on simulated thermal excitations to overcome potential barriers. Quantum annealing works in a similar way, employing quantum tunnneling to get through such barriers.
The objective function is equivalent to the Hamiltonian of the annealer, which is an ising model hamiltonian of the form: \[ \mathcal{H}(t) =\sum_{(i,j) \in E} J_{i,j} \sigma_i^z \sigma_j^z+ \sum_{i \in V} h_i \sigma_i^z+\Gamma(t) \sum_{i \in V} \sigma_i^x \\ \equiv \mathcal{H}_0-\Gamma(t) \sum_{i \in V} \sigma_i^x \] where the previous binary variable is now a spin variable as the hardware runs on superconducting qubits instead of transistors. More information on the nature of qubits can be found in the qubits section of this webpage, and a more detailed description of the ising model along with numerical results of simulated vs quantum annealing can be found in here.
Here the \(\sigma_i^z\) are the pauli spin operators and the spins can be measured to be \( \pm 1\). The \(J_{i,j} \) and \(h_i\) are the qubit coupling strengths and biases, and \( \Gamma \) is a time-varying transverse magnetic field. Coupling strength is controlled by couplers, which can be either ferromagnetic or anti-ferromagnetic, and quantifies how much nearest neighbour qubits may interact. More specifically, ferromagnetic coupling makes qubits want to align while the anti-ferromagnetic coulping makes them what to anti-align. Physically, this is known as entanglement and allows the qubits to form an entangled state which is the quantum wavefunction that the Hamiltonian \(\mathcal{H}(t)\) measures the energy of.
During the annealing process, the magnetic field starts off strong and is slowly reduced to zero. If this process is slow enough then if the system begins in its ground state, it should finish
in the ground state, hence providing the configuration of spins that minimises the energy of the problem hamiltonian, \( \mathcal{H}_0 \). This works due to the adiabatic theorem, which states that if a
closed quantum system described by a time dependent Hamiltonian \(\mathcal{H}(t)\) is initially prepared in its ground state, then it remains in its ground state provided the rate
of change of the Hamiltonian is sufficiently slow. More specifically, if the process is adiabatic (no heat goes in or out) and the time of the anneal T satisfies the adiabatic condition:
\[ T \gg \max _{s \in[0,1]} \frac{\left|\left\langle E_1|d H / d s| E_0\right\rangle\right|}{\Delta_{\mathrm{10}}^2} \]
where \(s\) is normalised time, \(t/T\) and the denominator is the modulus sqaured of the energy difference between the ground and 1st excited state energies. The value of \(\Delta_{\mathrm{10}}^2\) changes as \(\mathcal{H}(t)\) varies
since the energy eigenspectrum of the Hamiltonian changes over time, as shown in Figure 2.

Hence we arrive at another signifiant limiting factor of quantum annealing - the maximum of this fraction is at the minimum energy gap between the ground and 1st excited state, which can be very small. Finding this value is at least as hard as finding the ground state of \( \mathcal{H}_0 \) and hence difficulty arises in ensuring the process happens adiabatically. However, it's important to note that the first excited state is still low energy, just not the minimum energy, so if the adiabatic limit is not adhered to perfectly an approximate solution may still be found.
Industry Application - D-wave systems
D-wave Systems built and marketed their first superconducting-circuit Quantum Annealer in 2011 which had 128 qubits and 352 couplers. In 2020 they released their 'Advantage' processor, with 5640 super-conducing flux qubits and 40,484 couplers. They are leading the quantum annealing industry and making fast progress, but they still face the same afformentioned difficulties with problem limitations and time constraints. Their Hamiltonian takes a slightly different form: \[\mathcal{H}_{\text {ising }}=\underbrace{\frac{A(s)}{2}\left(\sum_{i \in V} \hat{\sigma}_i^{x}\right)}_{\text {Initial Hamiltonian }}+\underbrace{\frac{B(s)}{2}\left(\sum_{i \in V} h_i \hat{\sigma}_i^{z}+\sum_{(i,j) \in E} J_{i, j} \hat{\sigma}_i^{z} \hat{\sigma}_l^{z}\right)}_{\text {Final Hamiltonian }}\] where \( \mathcal{H}_0 \) is now simply superposition state of all the qubits in their ground state with equal probability of up and down spin projection, and the amplitude of both hamiltonians is controlled in tandem. The problem hamiltonian is slowly implemented through the use of controlling magnetic fields and introducing couplings as previously mentioned. This initial quantum state is comparitively easy to prepare and if the adiabatic condition is followed D-wave annealers should be able to find a good solution, although repeated runs may be required to sieve out runs where the state got too excited.
A signifiant obstacle facing D-wave systems is the process of embedding the graph representing the logical graph into the graph representing the arrangements of qubits. As in figure 1, the most recent annealer has a graph of degree 15, meaning 1 qubit connects to 15 others, which may not be sufficient to directly implement all the \(Q_{i,j}\) terms in the problem objective function. As such, mapping the equivalent problem requires the creation of chains of strongly coupled qubits which represent a single problem variable, as demonstrated in Figure 3. Such chains require strongly coupled qubits to not break during the anneal, which can interefere with the intra-chain couplings \(J_{i,j}\) if the magnitude of the coupling is too strong, and introduce errors. Minimsing the effect of this and making annealers more effective is an active area of research, for more read this paper. An interesting limitation is that if the graphs are hihly connected, couplings can interfere in an effect called noise aggregation, reducing the effectiveness of highly connected graphs which could have been embedded into graphs of a lesser degree.

Figure 3 - Embedding using chains of Qubits. Taken from this paper.
Quantum advantage?
Early numerical results found that simulated quantum annealing outperformed the already well-established
simulated thermal annealing when finding the ground state of small size systems. A strong theoretical argument for this is the thermal excitations used to escape local
minima are limited by the height of potential barriers while the quantum state may tunnel through arbritrarily high yet narrow barriers. This is to say that
quantum annealing is a more effective heuristic method at finding the global minimum of QUBO problems. Indeed in 2023, D-waves 'Advantage' processor outperformed
analogous monte carlo methods in finding the ground state of a 3D spin glass- an irregular 3D system of magnetic spins with random couplings.
While this is a significant step towards the ever sought-after 'quantum advantage' that quantum computing enthusiasts and academics have promised for decades, it sadly is not a definitive advantage.
The problem solved was very physical, so while this means annealers are useful machines for solving problems in physics, real-world problems are still yet to be favoured by the machines. There are a lot of speculative applications
of annealing across quantum chemistry, logistics problems in industry, finance and machine learning to name a few- all of which can be read about here. Interestingly, D-wave
annealers are already being tested on industry data- for example in 2019 D-wave's 2000Q annealer was used on airport data and successfully optimised flight gate assignment.
Hence, given the current state of quantum annealing and the continuous research efforts behind it, it is likely that industry effective Quantum Annealers are only a matter of time away.