Wydaje się, że obliczenia kwantowe są często rozumiane jako metoda obliczeń w obwodzie kwantowym, w której rejestr kubitów działa na obwód bramek kwantowych i jest mierzony na wyjściu (i ewentualnie na niektórych etapach pośrednich). Wyżarzanie kwantowe przynajmniej wydaje się być zupełnie inną metodą niż obliczanie zasobów kwantowych ¹ , ponieważ nie obejmuje bramek kwantowych.

Jakie są różne modele obliczeń kwantowych? Co ich wyróżnia?

Aby wyjaśnić, nie pytam, jakie różne kubityczne implementacje fizyczne mają na myśli, mam na myśli opis różnych pomysłów, jak obliczyć dane wyjściowe z danych wejściowych ² przy użyciu zasobów kwantowych.

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<sub>1. Wszystko, co z natury nie jest klasyczne, jak splątanie i spójność.
2. Proces, który przekształca dane wejściowe (takie jak kubity) na dane wyjściowe (wyniki obliczeń).</sub>

Model adiabatyczny

Ten model obliczeń kwantowych jest motywowany ideami kwantowej teorii wielu ciał i różni się zasadniczo zarówno od modelu obwodu (tym, że jest to model ciągły w czasie), jak i od ciągłych spacerów kwantowych (tym, że ma ewolucja zależna).

Obliczenia adiabatyczne zwykle przyjmują następującą postać.

1. Zacznij od zestawu kubitów, wszystkie w prostym stanie, takim jak . Wywołaj początkowy stan globalny | ψ 0 ⟩ .$\lvert + \rangle$$\lvert \psi_0 \rangle$
2. Poddaj te kubity interakcji Hamiltonian dla której | ψ 0 ⟩ jest wyjątkowy stan podstawowy (stan o najniższej energii). Na przykład dane | ψ 0 ⟩ = | + ⟩ ⊗ n , możemy wybrać H 0 = - ∑ k σ ( x ) k .$H_0$$\lvert \psi_0 \rangle$$\lvert \psi_0 \rangle = \lvert + \rangle^{\otimes n}$$H_0 = - \sum_{k} \sigma^{(x)}_k$
3. Wybierz końcowy hamiltonian , który ma unikalny stan podstawowy, który koduje odpowiedź na interesujący Cię problem. Na przykład, jeśli chcesz rozwiązać problem satysfakcji z ograniczeń, możesz zdefiniować hamiltonian H 1 = ∑ c h c , gdzie suma jest przejmowana przez ograniczenia c problemu klasycznego, a gdzie każdy h c jest operatorem, który nakłada karę energetyczną (dodatni wkład energii) na dowolny standardowy stan podstawowy reprezentujący klasyczne przypisanie, które nie spełnia ograniczenia c .$H_1$$H_1 = \sum_{c} h_c$$c$$h_c$$c$
4. Zdefiniuj przedział czasu i zmienny w czasie Hamiltonian H ( t ) tak, że H ( 0 ) = H 0 i H ( T ) = H 1 . Częstym, ale niekoniecznym wyborem jest po prostu przyjęcie interpolacji liniowej H ( t ) = $T \geqslant 0$$H(t)$$H(0) = H_0$$H(T) = H_1$.$H(t) = \tfrac{t}{T} H_1 + (1 - \tfrac{t}{T})H_0$
5. Dla czasów od do t = T pozwól układowi ewoluować pod ciągle zmieniającym się hamiltonianem H ( t ) i zmierz kubity na wyjściu, aby uzyskać wynik y ∈ { 0 , 1 } n .$t = 0$$t = T$$H(t)$$y \in \{0,1\}^n$

Podstawą modelu adiabatycznego jest twierdzenie adiabatyczne , którego jest kilka wersji. Wersja Ambainisa i Regeva [  arXiv: quant-ph / 0411152  ] (bardziej rygorystyczny przykład) implikuje, że jeśli zawsze istnieje „przerwa energetyczna” co najmniej między stanem podstawowym H ( t ) a jej pierwszym stan wzbudzony dla wszystkich 0 ⩽ t ⩽ T , a normy operatorowe pierwszej i drugiej pochodnej H są wystarczająco małe (to znaczy H ( t $\lambda > 0$$H(t)$$0 \leqslant t \leqslant T$$H$$H(t)$nie zmienia się zbyt szybko ani gwałtownie), wówczas możesz zwiększyć prawdopodobieństwo uzyskania żądanej wielkości wyjściowej tak szybko, jak chcesz, uruchamiając obliczenia wystarczająco wolno. Co więcej, możesz zmniejszyć prawdopodobieństwo błędu o dowolny stały czynnik, po prostu spowalniając całe obliczenia o czynnik związany z wielomianem.

Pomimo bardzo odmiennej prezentacji od modelu obwodu jednostkowego, wykazano, że model ten jest wielomianem równoważnym modelowi obwodu jednostkowego [  arXiv: quant-ph / 0405098  ]. Zaletą algorytmu adiabatycznego jest to, że zapewnia on inne podejście do konstruowania algorytmów kwantowych, które jest bardziej podatne na problemy z optymalizacją. Jedną wadą jest to, że nie jest jasne, jak zabezpieczyć go przed hałasem, ani powiedzieć, jak jego wydajność spada pod niedoskonałą kontrolą. Innym problemem jest to, że nawet bez niedoskonałości systemu określenie, jak wolno uruchomić algorytm, aby uzyskać wiarygodną odpowiedź, jest trudnym problemem - zależy od luki energetycznej i ogólnie nie jest łatwo powiedzieć, jaka energia przerwa jest dla statycznego hamiltonianu $H$, nie mówiąc już o zmiennym w czasie jednym .$H(t)$

Mimo to jest to model zarówno teoretyczny, jak i praktyczny, i wyróżnia się tym, że najbardziej różni się od modelu obwodu jednorodnego zasadniczo każdego z istniejących.

Obliczenia kwantowe oparte na pomiarach (MBQC)

Jest to sposób wykonywania obliczeń kwantowych, wykorzystujący pomiary pośrednie jako sposób napędzania obliczeń, a nie tylko wyodrębnianie odpowiedzi. Jest to szczególny przypadek „obwodów kwantowych z pomiarami pośrednimi”, a zatem nie ma już większej mocy. Jednak, kiedy został wprowadzony, podniósł intuicję wielu ludzi na temat roli przekształceń jednostkowych w obliczeniach kwantowych. W tym modelu istnieją ograniczenia, takie jak:

1. Przygotowuje się lub otrzymuje się bardzo duży stan splątany - taki, który można opisać (lub przygotować), mając pewien zestaw kubitów, wszystkie wstępnie przygotowane w tym stanie , A następnie pewna sekwencja operacji o kontrolowanym Z C Z = d i a g ( + 1 , + 1 , + 1 , - 1 ) , przeprowadzonych na parach kubitów zgodnie z relacjami krawędzi wykresu (zwykle prostokątna siatka lub sześciokątna sieć).$\lvert + \rangle$$\mathrm{CZ} = \mathrm{diag}(+1,+1,+1,-1)$
2. Wykonaj sekwencję pomiarów na tych kubitach - niektóre być może w standardowej podstawie, ale większość nie w standardowej podstawie, ale zamiast tego mierz obserwowalne wartości, takie jak dla różne kąty θ . Każdy pomiar daje wynik + 1 lub - 1 (często oznaczony odpowiednio jako „0” lub „1”), a wybór kąta może zależeć w prosty sposób od wyników poprzednich pomiarów (w sposób obliczony na podstawie klasycznego Układ sterowania).$M_{\mathrm{XY}}(\theta) = \cos(\theta) X - \sin(\theta) Y$$\theta$$+1$$-1$
3. Odpowiedź na obliczenia można obliczyć z klasycznych wyników pomiarów.$\pm 1$

Podobnie jak w przypadku modelu z obwodem jednolitym, istnieją warianty, które można rozważyć dla tego modelu. Jednak podstawową koncepcją są adaptacyjne pomiary pojedynczych kubitów wykonywane w dużym stanie splątania lub w stanie, który został poddany sekwencji operacji dojazdów i ewentualnie splątania, które są wykonywane jednocześnie lub etapami.

Ten model obliczeń jest zwykle uważany za przydatny przede wszystkim jako sposób symulacji obwodów jednostkowych. Ponieważ często jest postrzegany jako środek do symulacji lepiej lubianego i prostszego modelu obliczeń, dla większości ludzi nie jest już uważany za bardzo interesujący. Jednak:

• Jest to ważne między innymi jako motywująca koncepcja klasy IQP , która jest jednym ze sposobów wykazania, że ​​komputer kwantowy jest trudny do symulacji, oraz Blind Quantum Computing, który jest jednym ze sposobów rozwiązania problemów w bezpiecznych obliczeniach z wykorzystaniem zasobów kwantowych .

• Nie ma powodu, dla którego obliczenia oparte na pomiarach powinny być zasadniczo ograniczone do symulacji jednolitych obwodów kwantowych: wydaje mi się (i garstka innych teoretyków w mniejszości), że MQBC może zapewnić sposób opisywania interesujących pierwotnych obliczeń. Chociaż MBQC jest tylko specjalnym przypadkiem obwodów z pomiarami pośrednimi, a zatem może być symulowany przez obwody jednostkowe z jedynie wielomianem narzutu, nie oznacza to, że obwody jednolite byłyby koniecznie bardzo owocnym sposobem opisania wszystkiego, co można zrobić w zasadzie w obliczeniach opartych na pomiarach (tak jak istnieją imperatywne i funkcjonalne języki programowania w obliczeniach klasycznych, które są ze sobą trochę nieswojo).

Pozostaje pytanie, czy MBQC zasugeruje jakikolwiek sposób myślenia o budowie algorytmów, który nie jest tak łatwo prezentowany w kategoriach obwodów jednostkowych - ale nie może być mowy o przewadze obliczeniowej lub wady w stosunku do obwodów jednolitych, z wyjątkiem jednego o określonych zasobach i przydatności dla trochę architektury.

The Unitary Circuit Model

This is the best well-known model of quantum computation. In this model one has constraints such as the following:

1. $\lvert 0 \rangle$;
2. $x\in \{0,1\}^n$;
3. one or more measurements in the standard basis performed at the very end of the computation, yielding a classical output string $y \in \{0,1\}^k$. (We do not require $k = n$: for instance, for YES / NO problems, one often takes $k = 1$ no matter the size of $n$.)

Minor details may change (for instance, the set of unitaries one may perform; whether one allows preparation in other pure states such as $\lvert 1 \rangle$, $\lvert +\rangle$, $\lvert -\rangle$; whether measurements must be in the standard basis or can also be in some other basis), but these do not make any essential difference.

Discrete-time quantum walk

A "discrete-time quantum walk" is a quantum variation on a random walk, in which there is a 'walker' (or multiple 'walkers') which takes small steps in a graph (e.g. a chain of nodes, or a rectangular grid). The difference is that where a random walker takes a step in a randomly determined direction, a quantum walker takes a step in a direction determined by a quantum "coin" register, which at each step is "flipped" by a unitary transformation rather than changed by re-sampling a random variable. See [ arXiv:quant-ph/0012090 ] for an early reference.

For the sake of simplicity, I will describe a quantum walk on a cycle of size $2^n$; though one must change some of the details to consider quantum walks on more general graphs. In this model of computation, one typically does the following.

1. Prepare a "position" register on $n$ qubits in some state such as $\lvert 00\cdots 0\rangle$, and a "coin" register (with standard basis states which we denote by $\lvert +1 \rangle$ and $\lvert -1 \rangle$) in some initial state which may be a superposition of the two standard basis states.
2. Perform a coherent controlled-unitary transformation, which adds 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert +1 \rangle$, and subtracts 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert -1 \rangle$.
3. Perform a fixed unitary transformation $C$ to the coin register. This plays the role of a "coin flip" to determine the direction of the next step. We then return to step 2.

The main difference between this and a random walk is that the different possible "trajectories" of the walker are being performed coherently in superposition, so that they can destructively interfere. This leads to a walker behaviour which is more like ballistic motion than diffusion. Indeed, an early presentation of a model such as this was made by Feynmann, as a way to simulate the Dirac equation.

This model also often is described in terms of looking for or locating 'marked' elements in the graph, in which case one performs another step (to compute whether the node the walker is at is marked, and then to measure the outcome of that computation) before returning to Step 2. Other variations of this sort are reasonable.

To perform a quantum walk on a more general graph, one must replace the "position" register with one which can express all of the nodes of the graph, and the "coin" register with one which can express the edges incident to a vertex. The "coin operator" then must also be replaced with one which allows the walker to perform an interesting superposition of different trajectories. (What counts as 'interesting' depends on what your motivation is: physicists often consider ways in which changing the coin operator changes the evolution of the probability density, not for computational purposes but as a way of probing at basic physics using quantum walks as a reasonable toy model of particle movement.) A good framework for generalising quantum walks to more general graphs is the Szegedy formulation [ arXiv:quant-ph/0401053 ] of discrete-time quantum walks.

This model of computation is strictly speaking a special case of the unitary circuit model, but is motivated with very specific physical intuitions, which has led to some algorithmic insights (see e.g. [ arXiv:1302.3143 ]) for polynomial-time speedups in bounded-error quantum algorithms. This model is also a close relative of the continuous-time quantum walk as a model of computation.

Quantum circuits with intermediary measurements

This is a slight variation on "unitary circuits", in which one allows measurements in the middle of the algorithm as well as the end, and where one also allows future operations to depend on the outcomes of those measurements. It represents a realistic picture of a quantum processor which interacts with a classical control device, which among other things is the interface between the quantum processor and a human user.

Intermediary measurement is practically necessary to perform error correction, and so this is in principle a more realistic picture of quantum computation than the unitary circuit model. but it is not uncommon for theorists of a certain type to strongly prefer measurements to be left until the end (using the principle of deferred measurement to simulate any 'intermediary' measurements). So, this may be a significant distinction to make when talking about quantum algorithms — but this does not lead to a theoretical increase in the computational power of a quantum algorithm.

Quantum annealing

Quantum annealing is a model of quantum computation which, roughly speaking, generalises the adiabatic model of computation. It has attracted popular — and commercial — attention as a result of D-WAVE's work on the subject.

Precisely what quantum annealing consists of is not as well-defined as other models of computation, essentially because it is of more interest to quantum technologists than computer scientists. Broadly speaking, we can say that it is usually considered by people with the motivations of engineers, rather than the motivations of mathematicians, so that the subject appears to have many intuitions and rules of thumb but few 'formal' results. In fact, in an answer to my question about quantum annealing, Andrew O goes so far as to say that "quantum annealing can't be defined without considerations of algorithms and hardware". Nevertheless, "quantum annealing" seems is well-defined enough to be described as a way of approaching how to solve problems with quantum technologies with specific techniques — and so despite Andrew O's assessment, I think that it embodies some implicitly defined model of computation. I will attempt to describe that model here.

Intuition behind the model

Quantum annealing gets its name from a loose analogy to (classical) simulated annealing. They are both presented as means of minimising the energy of a system, expressed in the form of a Hamiltonian:

$$\begin{aligned} H_{\rm{classical}} &= \sum_{i,j} J_{ij} s_i s_j \\ H_{\rm{quantum}} &= A(t) \sum_{i,j} J_{ij} \sigma_i^z \sigma_j^z - B(t) \sum_i \sigma_i^x \end{aligned}$$ With simulated annealing, one essentially performs a random walk on the possible assignments to the 'local' variables $s_i \in \{0,1\}$, but where the probability of actually making a transition depends on

• The difference in 'energy' $\Delta E = E_1 - E_0$ between two 'configurations' (the initial and the final global assignment to the variables $\{s_i\}_{i=1}^n$) before and after each step of the walk;
• A 'temperature' parameter which governs the probability with which the walk is allowed to perform a step in the random walk which has $\Delta E > 0$.

One starts with the system at 'infinite temperature', which is ultimately a fancy way of saying that you allow for all possible transitions, regardless of increases or decreases in energy. You then lower the temperature according to some schedule, so that time goes on, changes in state which increase the energy become less and less likely (though still possible). The limit is zero temperature, in which any transition which decreases energy is allowed, but any transition which increases energy is simply forbidden. For any temperature $T > 0$, there will be a stable distribution (a 'thermal state') of assignments, which is the uniform distribution at 'infinite' temperature, and which is which is more and more weighted on the global minimum energy states as the temperature decreases. If you take long enough to decrease the temperature from infinite to near zero, you should in principle be guaranteed to find a global optimum to the problem of minimising the energy. Thus simulated annealing is an approach to solving optimisation problems.

Quantum annealing is motivated by generalising the work by Farhi et al. on adiabatic quantum computation [arXiv:quant-ph/0001106], with the idea of considering what evolution occurs when one does not necessarily evolve the Hamiltonian in the adiabatic regime. Similarly to classical annealing, one starts in a configuration in which "classical assignments" to some problem are in a uniform distribution, though this time in coherent superposition instead of a probability distribution: this is achieved for time $t = 0$, for instance, by setting

$$A(t=0) = 0, \qquad B(t=0) = 1$$ in which case the uniform superposition $\def\ket#1{\lvert#1\rangle}\ket{\psi_0} \propto \ket{00\cdots00} + \ket{00\cdots01} + \cdots + \ket{11\cdots11}$ is a minimum-energy state of the quantum Hamiltonian. One steers this 'distribution' (i.e. the state of the quantum system) to one which is heavily weighted on a low-energy configuration by slowly evolving the system — by slowly changing the field strengths $A(t)$ and $B(t)$ to some final value $$A(t_f) = 1, \qquad B(t_f) = 0.$$ Again, if you do this slowly enough, you will succeed with high probability in obtaining such a global minimum. The adiabatic regime describes conditions which are sufficient for this to occur, by virtue of remaining in (a state which is very close to) the ground state of the Hamiltonian at all intermediate times. However, it is considered possible that one can evolve the system faster than this and still achieve a high probability of success.

Similarly to adiabatic quantum computing, the way that $A(t)$ and $B(t)$ are defined are often presented as a linear interpolations from $0$ to $1$ (increasing for $A(t)$, and decreasing for $B(t)$). However, also in common with adiabatic computation, $A(t)$ and $B(t)$ don't necessarily have to be linear or even monotonic. For instance, D-Wave has considered the advantages of pausing the annealing schedule and 'backwards anneals'.

'Proper' quantum annealing (so to speak) presupposes that evolution is probably not being done in the adiabatic regime, and allows for the possibility of diabatic transitions, but only asks for a high chance of achieving an optimum — or even more pragmatically still, of achieving a result which would be difficult to find using classical techniques. There are no formal results about how quickly you can change your Hamiltonian to achieve this: the subject appears mostly to consist of experimenting with a heuristic to see what works in practise.

The comparison with classical simulated annealing

Despite the terminology, it is not immediately clear that there is much which quantum annealing has in common with classical annealing. The main differences between quantum annealing and classical simulated annealing appear to be that:

• In quantum annealing, the state is in some sense ideally a pure state, rather than a mixed state (corresponding to the probability distribution in classical annealing);

• In quantum annealing, the evolution is driven by an explicit change in the Hamiltonian rather than an external parameter.

It is possible that a change in presentation could make the analogy between quantum annealing and classical annealing tighter. For instance, one could incorporate the temperature parameter into the spin Hamiltonian for classical annealing, by writing

$$\tilde H_{\rm{classical}} = A(t) \sum_{i,j} J_{ij} s_i s_j - B(t) \sum_{i,j} \textit{const.}$$ where we might choose something like $A(t) = t\big/(t_F - t)$ and $B(t) = t_F - t$ for $t_F > 0$ the length of the anneal schedule. (This is chosen deliberately so that $A(0) = 0$ and $A(t) \to +\infty$ for $t \to t_F$.) Then, just as an annealing algorithm is governed in principle by the Schrödinger equation for all times, we may consider an annealing process which is governed by a diffusion process which is in principle uniform with tim by small changes in configurations, where the probability of executing a randomly selected change of configuration is governed by $$p(x \to y) = \max\Bigl\{ 1,\; \exp\bigl(-\gamma \Delta E_{x\to y}\bigr) \Bigr\}$$ for some constant $\gamma$, where $E_{x \to y}$ is the energy difference between the initial and final configurations. The stable distribution of this diffusion for the Hamiltonian at $t=0$ is the uniform distribution, and the stable distribution for the Hamiltonian as $t \to t_F$ is any local minimum; and as $t$ increases, the probability with which a transition occurs which increases the energy becomes smaller, until as $t \to t_F$ the probability of any increases in energy vanish (because any of the possible increase is a costly one).

There are still disanalogies to quantum annealing in this — for instance, we achieve the strong suppression of increases in energy as $t \to t_F$ essentially by making the potential wells infinitely deep (which is not a very physical thing to do) — but this does illustrate something of a commonality between the two models, with the main distinction being not so much the evolution of the Hamiltonian as it is the difference between diffusion and Schrödinger dynamics. This suggests that there may be a sharper way to compare the two models theoretically: by describing the difference between classical and quantum annealing, as being analogous to the difference between random walks and quantum walks. A common idiom in describing quantum annealing is to speak of 'tunnelling' through energy barriers — this is certainly pertinent to how people consider quantum walks: consider for instance the work by Farhi et al. on continuous-time quantum speed-ups for evaluating NAND circuits, and more directly foundational work by Wong on quantum walks on the line tunnelling through potential barriers. Some work has been done by Chancellor [arXiv:1606.06800] on considering quantum annealing in terms of quantum walks, though it appears that there is room for a more formal and complete account.

On a purely operational level, it appears that quantum annealing gives a performance advantage over classical annealing (see for example these slides on the difference in performance between quantum vs. classical annealing, from Troyer's group at ETH, ca. 2014).

Quantum annealing as a phenomenon, as opposed to a computational model

Because quantum annealing is more studied by technologists, they focus on the concept of realising quantum annealing as an effect rather than defining the model in terms of general principles. (A rough analogy would be studying the unitary circuit model only inasmuch as it represents a means of achieving the 'effects' of eigenvalue estimation or amplitude amplification.)

Therefore, whether something counts as "quantum annealing" is described by at least some people as being hardware-dependent, and even input-dependent: for instance, on the layout of the qubits, the noise levels of the machine. It seems that even trying to approach the adiabatic regime will prevent you from achieving quantum annealing, because the idea of what quantum annealing even consists of includes the idea that noise (such as decoherence) will prevent annealing from being realised: as a computational effect, as opposed to a computational model, quantum annealing essentially requires that the annealing schedule is shorter than the decoherence time of the quantum system.

Some people occasionally describe noise as being somehow essential to the process of quantum annealing. For instance, Boixo et al. [arXiv:1304.4595] write

Unlike adiabatic quantum computing[, quantum annealing] is a positive temperature method involving an open quantum system coupled to a thermal bath.

It might perhaps be accurate to describe it as being an inevitable feature of systems in which one will perform annealing (just because noise is inevitable feature of a system in which you will do quantum information processing of any kind): as Andrew O writes "in reality no baths really help quantum annealing". It is possible that a dissipative process can help quantum annealing by helping the system build population on lower-energy states (as suggested by work by Amin et al., [arXiv:cond-mat/0609332]), but this seems essentially to be a classical effect, and would inherently require a quiet low-temperature environment rather than 'the presence of noise'.

The bottom line

It might be said — in particular by those who study it — that quantum annealing is an effect, rather than a model of computation. A "quantum annealer" would then be best understood as "a machine which realises the effect of quantum annealing", rather than a machine which attempts to embody a model of computation which is known as 'quantum annealing'. However, the same might be said of adiabatic quantum computation, which is — in my opinion correctly — described as a model of computation in its own right.

Perhaps it would be fair to describe quantum annealing as an approach to realising a very general heuristic, and that there is an implicit model of computation which could be characterised as the conditions under which we could expect this heuristic to be successful. If we consider quantum annealing this way, it would be a model which includes the adiabatic regime (with zero-noise) as a special case, but it may in principle be more general.