Continuous-time quantum algorithms for unstructured problems

The paradigm of adiabatic quantum computation (AQC) proposed by Farhi et al [1] about a decade ago is a simple yet intriguing approach to problem solving on a quantum computer. Unlike the leading paradigm of circuit-based quantum computing, AQC is an analogue continuous-time method that does not require the design and use of quantum gates. As such, it can in many ways be thought of as a simpler and perhaps more profound method for performing quantum computations that is also easier to implement experimentally [2, 3].

Even though AQC has been shown to be polynomially-equivalent to circuit-based computation [4, 5], and despite intensive research in the area (see, e.g., [6–14] and references therein), to date, there are almost no clear-cut concrete examples for efficient quantum adiabatic algorithms (QAAs) that reveal the potentially-powerful 'fully-quantum' capabilities encompassed in AQC. One possible reason for that is, presumably, that there is usually no obvious way to 'tailor' the adiabatic algorithm to the specific problem being examined, and to make use of the structure of the problem to, for example, modify the beginning Hamiltonian in a clever way that would speed up the computation (a notable exception is [15]). For most of the interesting optimization problems, being able to do so, may be as hard as solving the original problem itself¹.

Interestingly, one of the few problems for which quantum speedup has been obtained in the context of AQC, is Grover's unstructured search problem [17], in which one searches for a marked item in an unstructured database. Roland and Cerf [18] have demonstrated that while the application of the adiabatic algorithm to Grover's problem with a linear rate results in a running time that is of order N, where N is the number of items in the database, a carefully chosen variable rate of the adiabatic parameter yields a running time that scales as $\sqrt{N}$, i.e., a quadratic speed-up is gained, similarly to the original result by Grover found for the circuit-based model².

The adiabatic algorithm for Grover's problem utilizes the concept of local adiabatic evolution, in the framework of which the adiabatic parameter is varied not at a constant rate but rather at a variable rate, slowing down in the vicinity of the minimum gap and speeding up in places where the gap is large.

Local adiabatic evolution however can only be useful in cases where one has proper knowledge of the exact behavior of the gap and relevant matrix elements of the system for the problem in question. This is normally not the case. In the Grover problem, the ability to compute the gap and matrix element of the problem stems from prior knowledge of the spectrum of the problem Hamiltonian, which, ultimately, reduces the problem into a simple two-level system [18].

In what follows, we consider a family of unstructured optimization problems, which we refer to as 'scrambled input' models, and show how one may utilize knowledge of the spectrum of the Hamiltonian of the problem, to find analogue, continuous-time (i.e., not necessarily adiabatic) algorithms that are faster than their classical analogues. In that sense, this family of problems may be viewed as a generalization of, or an extension to, the problem solved by Roland and Cerf [18].

In scrambled input problems, one has to find a minimum input configuration (i.e., $\arg \min$) of an n-bit function whose set of outputs (and their multiplicities) is given in advance, up to an unknown constant offset. The exact mapping between the 2ⁿ input configurations and the various outputs is also not given (i.e., it is as if the inputs of a known function have been scrambled in an unknown way). As we also discuss later, special cases in this family of problems are the unstructured database search problem considered by Roland and Cerf [18], certain variants of the random energy model and the functions of the Deutsch–Josza problem.

We illustrate the manner in which AQC may be used to solve scrambled problems by constructing efficient analogue continuous-time algorithms for two specific examples: the Deutsch–Josza [20] problem for which we find an efficient O(1) non-adiabatic deterministic algorithm, and a variant of the random energy model, for which the minimum energy configuration is found quadratically faster than the corresponding classical algorithms.

The paper is organized as follows. In the next section we briefly discuss the principles of the QAA that is the heart of AQC, and with which the above models are solved. In section 3, we describe scrambled input problems in detail. We then study two examples. In section 4, we suggest an analogue algorithm for the Deutsch–Josza problem, and in section 5 we consider a variant of the random energy model. Finally, we conclude with a few comments in section 6.

The QAA provides the general approach for solving optimization problems on an analogue continuous-time quantum computer [1]. Within the framework of the QAA, the solution to an optimization problem is encoded in the ground state of a Hamiltonian $\hat{H}_p$. To find the solution, the QAA prescribes the following course of action. As a first step, the system is prepared in the ground state of another 'driver' Hamiltonian $\hat{H}_d$. The driver Hamiltonian is chosen such that it does not commute with the problem Hamiltonian and has a ground state that is fairly easy to prepare. As a next step, the Hamiltonian of the system is slowly modified from $\hat{H}_d$ to $\hat{H}_p$, using the linear interpolation, i.e.,

$$\begin{equation} \hat{H}(s)=s \hat{H}_p +(1-s) \hat{H}_d , \end{equation} \tag{ 1 }$$

where s(t) is a parameter varying smoothly with time, from 0 at t = 0 to 1 at the end of the algorithm, $t=\mathcal {T}$. If this process is done slowly enough, the adiabatic theorem of quantum mechanics (see, e.g., [21] and [22]) ensures that the system will stay close to the ground state of the instantaneous Hamiltonian throughout the evolution, so that one finally obtains a state close to the ground state of $\hat{H}_p$. At this point, measuring the state will give the solution of the original problem with high probability.

For adiabatic processes, the adiabatic profile function s(t) must be chosen such that the evolution of the system is slow. In the simple case where s(t) varies from zero to one at a constant rate, the runtime $\mathcal {T}$ must be chosen to be large enough so that the adiabatic approximation holds: this condition determines the efficiency, or complexity, of the QAA. A condition³ on $\mathcal {T}$ can be given in terms of the instantaneous eigenstates {|m〉} and eigenvalues {E_m} of the Hamiltonian $\hat{H}(s)$, as [7, 26]

$$\begin{equation} \mathcal {T} =\frac{1}{\epsilon } \, {\max _{s} V_{01}(s) \over \min _s g^2(s)} , \end{equation} \tag{ 2 }$$

where g(s) is the first excitation gap E₁(s) − E₀(s) and $V_{01}(s) = | \langle 0 | \mathrm{d}\hat{H}/ \mathrm{d}s | 1\rangle |$ (in our units ℏ = 1). Here, is a small number inversely proportional to the running time of the algorithm. The smaller is chosen to be, the slower the evolution of the system will be and the larger the probability of success will become. For an adiabatic process to have zero error, the running time must tend to infinity.

As discussed in the introduction, one could gain significant speed-up in calculation times in cases where the concept of local adiabatic evolution [18] can be utilized. This is done by formulating a 'local' Landau–Zener condition for each value of the adiabatic profile function s(t), namely:

$$\begin{equation} \left| \frac{\mathrm{d}s }{\mathrm{d}t} \right| \le \epsilon \frac{g^2(s)}{V_{01}(s)} , \end{equation} \tag{ 3 }$$

where , g(s) and V₀₁(s) are as in (2).

Before moving on to describe scrambled input problems, we note here that in special cases, the 'machinery' of the QAA may also be used differently than described above, namely, in a not-necessarily adiabatic fashion. By 'non-adiabatic' use, it is meant that the state of the system may 'wander away', at least to some extent, from the instantaneous ground-state of the system in the course of the algorithm before returning to it toward the end of the run. One example for a 'non-traditional' evolution (in which case the Hamiltonian is constant and the squared-overlap between the ground state and the instantaneous state of the system is constant) can be found in the analogue version of Grover's search algorithm suggested by Farhi and Gutmann in [8]. Another example, in which the state of the system does wander away from the instantaneous ground-state, is discussed in [27]. This concept is discussed later in greater detail.

In what follows, we consider a family of optimization problems, that we refer to as 'scrambled input' problems, and which, as we illustrate next, can be solved faster on an analogue quantum computer than on a classical computer. As we explain in what follows, this family of problems is a generalization of, or an extension to, the unstructured database search problem considered by Roland and Cerf [18].

In 'scrambled input' problems, we are asked to find the minimum cost and a minimizing configuration of an integer-valued n-bit function f( · ) that has the form

$$\begin{equation} f(x)=g(\pi (x))+e_0 . \end{equation} \tag{ 4 }$$

Here, x is an n-bit input and g( · ) is a known, given integer-valued n-bit function. The unknowns of the problem are π( · ) which is a permutation that 'scrambles' the inputs, and e₀—an unknown constant offset that is added to the outputs of g. For convenience, we denote the (distinct) known output values of g( · ) by g₀ < g₁ < ⋅⋅⋅ < g_K and their multiplicities by m₀, m₁, ..., m_K.⁴ Note that the output values of f( · ) are known up to the additive constant e₀, but the mapping between inputs and outputs is not. Our task is therefore to find a configuration x_* for which f(x_*) = f₀ ≡ g₀ + e₀.

Special cases in this family of problems are: (i) The original Grover's search problem considered by Roland and Cerf [18] in which g₀ = 0 with multiplicity m₀ = M (where M is the number of marked items or 'targets') and g₁ = 1 with m₁ = 2ⁿ − M. (ii) The Deutsch–Josza input-functions (which we discuss in the next section) for which both the balanced and constant functions are described by g₀ = 0 and g₁ = 1, with multiplicities m₀ = m₁ = 2^{n − 1} (balanced) and m_0/1 = 0 and m_1/0 = 2ⁿ (constant). (iii) A variant of the 'random energy model' for which g_j = j with j = 0, ..., n and $m_j={\big(\begin{array}{@{}c@{}} {n} {j}\end{array}\big)}$ (n-choose-j). This model has been previously considered in [8] in the context of the QAA.

Since the input values of the given function are scrambled in an unknown way, the typical running time of classical search algorithms designed to find a minimum-cost configuration will scale as O(2ⁿ) (given that the number of minimizing configurations is not exponential). Classically, the function f( · ) is called with one input at a time. If a call to the function yields an output that is not a minimum cost solution, the output will not contain any information as to the locations of minimizing configurations, due to the scrambling. For example, a classical deterministic solution to the problem will require in the worst case 2ⁿ − m₀ evaluations of the scrambled-input function.

In quantum-algorithmic terminology, the problem described above can be encoded in a matrix $\hat{F}$ that is diagonal in the computational basis and whose diagonal elements are

$$\begin{equation} \langle x | \hat{F} | x \rangle =f(x) , \end{equation} \tag{ 5 }$$

where |x〉 is the computational-basis state that corresponds to the classical input x. We will take $\hat{F}$ to be the problem Hamiltonian of the QAA procedure: $\hat{H}_p=\hat{F}$.

The lack of structure exhibited by the problem Hamiltonian above does not seem to allow for any clever 'tailoring' of the QAA to the problem, as no information about the problem can be incorporated into the driver Hamiltonian that would help in speeding up the computation. Therefore, a driver Hamiltonian that treats all computational-basis eigenvectors in exactly the same way will presumably be an optimal choice. We thus take as driver Hamiltonian, the one-dimensional projection onto the equal-superposition state:

$$\begin{equation} \hat{H}_d = - E_0 | \phi \rangle \langle \phi | , \end{equation} \tag{ 6 }$$

where E₀ is a positive constant that provides a scale to the driver Hamiltonian and |ϕ〉 is the fully-symmetric state

$$\begin{equation} | \phi \rangle = 2^{-n/2} \sum _{x=0}^{2^n-1} | x \rangle . \end{equation} \tag{ 7 }$$

The ground-state of this driver Hamiltonian, i.e., the state that the system is prepared in, for the adiabatic evolution, is

$$\begin{equation} |\psi (t=0) \rangle = | \phi \rangle , \end{equation} \tag{ 8 }$$

whose energy is −E₀.

Both the scrambled problem Hamiltonian and the above driver Hamiltonian have been shown to pose severe limitations on the complexity of any QAA constructed using either of the two [8]. We shall address this matter later on. Here, however, the above choice of driver Hamiltonian has an attractive property, at least as far as scrambled problem Hamiltonians are concerned. Its symmetry makes it invariant under any permutation of the eigenstates of the computational (problem Hamiltonian) basis. Specifically, it is invariant under the (unknown) unitary transformation that 'unscrambles', i.e., orders, the eigenvalues of the problem Hamiltonian. It thus follows that the Schrödinger equation of the QAA for this type of problems can be written down explicitly, and subsequently solved, at least in principle. This property of the system can then be used, as we show next, to find a (not necessarily) adiabatic path s(t) such that at the end of the algorithm, the final state of the system lies very close to (or in some cases, is precisely) the ground-state of the problem Hamiltonian. Once an optimal path is found, a measurement of the energy at the end of the run, will immediately reveal the minimum energy f₀ = e₀ + g₀ (and therefore also the value of the unknown offset e₀), along with the corresponding minimum-energy configuration.

Explicitly, the Schrödinger equation for a QAA on a scrambled problem, $\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t} |\psi (t)\rangle =\hat{H} |\psi (t) \rangle$, simplifies to the (K + 1) coupled first-order ordinary differential equations:

$$\begin{equation} \mathrm{i}\dot{c}_j = -(1-s) E_0 \sum _{k=0}^K \eta _k c_k + s g_j c_j , \end{equation} \tag{ 9 }$$

where j = 0, ..., K. Here, η_j = m_j/2ⁿ are the ratios of the multiplicities of occurrences of the jth eigenvalue g_j to the total number of eigenvalues 2ⁿ. The various c_j are the (K + 1) distinct wave-function amplitudes that correspond to the different values g_j. We have thus ended up with a compact set of equations which needs to be solved or analyzed in some meaningful way.

A few remarks are now in order: (i) Note that the effective dimension of the problem is K + 1, i.e., precisely the number of distinct g_j's. (ii) In terms of the different amplitudes c_j, the initial-state of the system corresponds to c_j(t = 0) = 1 for all j, where the normalization chosen here is $\sum _{j=0}^K \eta _j |c_j|^2=1$. Note that the evolving wave-function depends on n, the size of the problem, only through the various ratios. (iii) The addition of an unknown offset e₀ to the various g_j only affects the (immaterial) global phase of the wave-function, and may therefore be completely removed from the equations. On the other hand, it makes the minimum-energy configuration harder to find for classical algorithms.

In what follows, we utilize the ideas presented above, to obtain analogue (adiabatic, as well as non-adiabatic) algorithms for two exemplary problems for which no efficient classical solution is known. We first consider the famous Deutsch–Josza problem [20], for which we find an efficient constant-runtime deterministic solution, that stands in contrast to the exponential complexity of the corresponding deterministic classical algorithm, and is on par with the circuit-based quantum Deutsch–Josza algorithm. We next consider a variant of the random energy model already mentioned in the previous section, for which the minimum energy configuration is found quadratically faster than on a classical computer.

In the Deutsch–Josza problem, we are given a black box quantum computer (an oracle) that implements a boolean function of n bits that is either constant, i.e., gives 0 on all inputs or 1 on all inputs, or balanced, i.e., returns 1 for exactly half of the input states and 0 for the other half. The task then is to determine with absolute certainty whether the function is constant or balanced with as few calls as possible to the oracle.

While classical probabilistic algorithms solve the Deutsch–Josza problem with O(1) operations (i.e., independently of the size of the problem), a classical deterministic algorithm would require 2^{n − 1} + 1 evaluations of the function in the worst case (and two evaluations in the best case). The seminal result by Deutsch and Josza [20] was the construction of a deterministic circuit-based quantum algorithm that requires only two calls to the oracle (later reduced to only one call by Cleve et al [28]), thus providing the first example of a quantum computation that is exponentially faster than the best corresponding classical algorithm.

While the Deutsch–Josza problem does have the form of a 'scrambled input' problem as described above, in what follows we nonetheless show that one can use the principles discussed in the previous section to find an analogue continuous-time deterministic quantum computation to solve the Deutsch–Josza problem with a runtime that is on par with the analogous circuit-based algorithm, i.e., a runtime that does not scale with input size, and so is exponentially faster than the corresponding deterministic classical algorithm. We note here that probabilistic quantum adiabatic Deutsch–Josza algorithms already exist [29, 30], however, as already mentioned, these do not provide quantum speedups when compared against their classical counterpart.

The algorithm we propose here is very similar to adiabatic algorithms of the usual AQC paradigm but with one important distinction. While with adiabatic processes the success of the algorithm depends on the state of the system being close at all times to the instantaneous ground state of the evolving Hamiltonian, here shall 'relax' this condition. We will show that by allowing the state of the system to 'detach' itself at least to some extent from the instantaneous ground state, one can in fact obtain a deterministic (i.e., zero probability of failure) non-adiabatic efficient algorithms, for the Deutsch–Josza and certain other problems (see, e.g., [19]).

4.1. Dynamics of the Deutsch–Josza problem

In the context of scrambled problems, the Deutsch–Josza oracle corresponds in the 'constant' case to having g₀ = 0 or g₀ = 1 (with m₀ = 2ⁿ), and g₀ = 0 and g₁ = 1 with multiplicities m₀ = m₁ = 2^{n − 1} in the balanced case. The task here is to be able to distinguish between the two cases (and not to find the minimum of g( · )). In this problem, we shall assume, for simplicity, that the constant offset e₀ is zero (even though in principle it could take on any value).

To find a deterministic efficient solution to the Deutsch–Josza problem, we construct a continuous-time algorithm that is optimized for the case where the input matrix $\hat{F}$ is balanced, for reasons that will become clear shortly. In the balanced case, the problem Hamiltonian has two distinct values and the wave function |ψ(t)〉 has only two distinct components. It may therefore be written as

$$\begin{equation} |\psi (t) \rangle = c_0(t) | \psi _0\rangle + c_1(t) | \psi _1 \rangle , \end{equation} \tag{ 10 }$$

where |ψ₀〉 and |ψ₁〉 are the equal superpositions of all zero-energy states and states with energy one, respectively:

$$\begin{equation} |\psi _0 \rangle \equiv 2^{-(n-1)/2} \sum _{m \in \mathcal {M}} | m\rangle , \end{equation} \tag{ 11a }$$

$$\begin{equation} |\psi _1 \rangle \equiv 2^{-(n-1)/2} \sum _{m \notin \mathcal {M}} | m\rangle , \end{equation} \tag{ 11b }$$

where the 2^{n − 1} states $m \in \mathcal {M}$ are eigenstates of the computational basis with zero eigenvalues (all other states have eigenvalue one). The Schrödinger equation (9), thus becomes the two coupled equations (here, we fix the energy scale of the driver Hamiltonian at E₀ = 1):

$$\begin{equation} \mathrm{i}\dot{c}_0 = -\case{1}{2}(1-s) (c_0+c_1) \end{equation} \tag{ 12a }$$

$$\begin{equation} \mathrm{i}\dot{c}_1 = -\case{1}{2}(1-s) (c_0+c_1) +s c_1 . \end{equation} \tag{ 12b }$$

As previously mentioned, note that n, the size of the problem, does not appear in the equation.

The time-dependence of c₀ and c₁ is easily transformed into an s dependence. The time-derivative transforms in the usual way: d/dt = (ds/dt) × d/ds, and the two coupled equations now become one, with the new complex-valued dependent variable r(s) = c₁(s)/c₀(s):

$$\begin{equation} 2 \mathrm{i}r^{\prime }(s) = t^{\prime }(s)[ -1+s +2 s r(s) +(1 - s) r(s)^2] , \end{equation} \tag{ 13 }$$

with the initial condition of r(0) = 1 corresponding to the fully-symmetric initial state. Here, the prime symbol (') stands for differentiation with respect to the new independent variable s and t'(s) = 1/(ds/dt) encodes the non-adiabatic path.

As a next step, we split the above equation into its real and imaginary parts, re-expressing the complex-valued r(s) by real-valued components of magnitude and phase:

$$\begin{equation} r(s)=\sqrt{ \frac{1-p(s)}{p(s)}}\,\mathrm{e}^{\mathrm{i}\phi (s)} . \end{equation} \tag{ 14 }$$

Here, p(s) is the probability of the system to be in the symmetric zero-energy state |ψ₀〉 (11a) at any given s, and ϕ(s) is the phase, which is easily solved for, resulting in a single real-valued equation for the probability p(s).

A deterministic algorithm whose end-state is precisely the equal superposition of the zero-energy states, |ψ₀〉, may be obtained if we satisfy the requirement that the final state of the system at s = 1 will be precisely p(1) = 1 (equivalently, r(1) = 0). To make it so, we shall treat the Schrödinger equation, in this example (13), as an equation on the 'path' t'(s) rather than on the wave function itself (in our case, p(s)). Put differently, we first choose the wave function (r(s) here) or rather, the probability profile p(s), such that the probability of obtaining the desired solution is exactly one at the end of the run, and only then obtain the non-adiabatic path t'(s) consistently with the Schrödinger equation of the system (13).

For the Deutsch–Josza problem, an explicit expression for the path t'(s) as a function of the probability profile p(s) may be obtained:

$$\begin{equation} t^{\prime }(s) = \frac{p^{\prime }(s)}{\sqrt{p(s)(1-p(s))} \sqrt{1-q(s)^2}} , \end{equation} \tag{ 15 }$$

where

$$\begin{equation} q(s) = \frac{1-s[1+2 p(s)] + 2(1-s) \int _0^s \frac{p(^{\prime }s)}{(1-s^{\prime })^2} \,\mathrm{d}s^{\prime }}{2 (1-s) \sqrt{p(s)(1-p(s))}} . \end{equation} \tag{ 16 }$$

Presumably, there are infinitely many choices for p(s) that yield a desired path. One however must make certain that p(s) satisfies the boundary conditions, p(0) = 1/2 (corresponding to r(0) = 1), and p(1) = 1 (corresponding to r(1) = 0), and that the profile t'(s) is well-defined everywhere and yields a finite running time for the algorithm, namely, that $\mathcal {T}=\int _0^1 t^{\prime }(s) \,\mathrm{d}s < \infty$. One example for such a probability profile is

$$\begin{equation} p_*(s)=\case{1}{2}(1+6 s^2 - 8 s^3 + 3 s^4) , \end{equation} \tag{ 17 }$$

which immediately leads to

$$\begin{equation} t_*^{\prime }(s)=\frac{6 \sqrt{2}}{\sqrt{s (1-s)[4-9s(1-s)]}} . \end{equation} \tag{ 18 }$$

Figure 1 shows the 'adiabatic' parameter s_*(t) for the above example, obtained by integration and inversion of $t^{\prime }_*(s)$ above. The inset shows p_*(t), the generating probability profile.

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In figure 2, the probability of being in the excited-states superposition, namely, 1 − p_*(s), is plotted as a function of s on a logarithmic scale. As can be seen in the figure, this probability (solid line) drops sharply to zero as s approaches one. This is in contrast with the probability of failure in the adiabatic case (dashed line) generated by the adiabatic path t'(s) = [(1 − 2s(1 − s))]⁻¹ which does not vanish as s = 1 (here, = 0.001). Much may be learned by looking at the probability for the system to be in the instantaneous ground state throughout the evolution, P_IGS, as a function of s. In figure 3, this probability is shown both for the non-adiabatic case at hand (solid line) and also for the adiabatic case mentioned above (dashed). While for the latter choice the state of the system remains very close to the instantaneous ground state throughout the evolution, the 'deterministic' path slightly diverges from the adiabatic one before returning to it towards the end of the evolution.

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As mentioned above, there are presumably many paths s(t) that produce, for a carefully determined runtime, the desired end probability of one. We note here, that even the simple choice of s(t) = 1/2, while not conforming to the boundary conditions of s(0) = 1 and s(1) = 1, will also produce the desired probability. Another such path is $s(t)=t/\mathcal {T}_*$ where $\mathcal {T}_*$ is an appropriately chosen runtime.

4.2. The algorithm

In the following example, we consider a different model, namely, a variant of the random energy model [31, 32], previously considered under somewhat different settings in [33, 34] in the context of the QAA. Here, we show that one can utilize the knowledge of the spectrum of the problem Hamiltonian, to obtain an optimal adiabatic path that in turn yields a QAA to find the minimum-energy configuration. The resulting algorithm is found to be, as expected, quadratically-faster than any classical search algorithm.

In this variant of the random energy model, the eigenvalues of the problem Hamiltonian are integers in the range j ∈ {0, ..., n} where each value j appears precisely $m_j={\big(\begin{array}{@{}c@{}} {n} {j}\end{array}\big)}$ times. The 2ⁿ eigenvalues are distributed randomly along the diagonal. As discussed earlier, one could further complicate the problem by considering the addition of an arbitrary constant offset to the function values. Here too, one has to construct an algorithm designed to find the minimum-energy bit configuration (i.e., the configuration that corresponds to j = 0), and to determine how the running time of the algorithm scales with the dimension of the Hilbert space of the problem, 2ⁿ. Note that for the corresponding classical algorithm, the average running time would scale like O(2ⁿ), as the ground state has multiplicity one and there is no faster searching routine to find it, due to the lack of structure of the problem.

For what follows, we choose the energy scale of the driver Hamiltonian to be E₀ = n for computational convenience. The Schrödinger equation (9), becomes for this problem, a set of only (n + 1) linear first-order differential equations, corresponding to the (n + 1) distinct energies of the problem Hamiltonian.

Noting that the Hamiltonian is of the form $\hat{A}+ | u\rangle \langle u|$ where $\hat{A}$ is a diagonal matrix and |u〉 is a vector, one may use the matrix determinant lemma [35] to obtain the characteristic polynomial of the reduced Hamiltonian as a function of the adiabatic parameter s:

$$\begin{equation} \frac{n(1-s)}{2^n}\sum _{j=0}^{n} {\left(\begin{array}{@{}c@{}}n \\ {j}\end{array}\right)}\frac{1}{j s-\lambda }=1 . \end{equation} \tag{ 19 }$$

The solutions λ to this equation are the energy levels of the Hamiltonian, and may be obtained analytically in the large n limit.

Figure 4 shows the three lowest energy levels of the system as a function of s for n = 20 (the energy levels are qualitatively similar for larger all n values as well). Note that the gap between the ground state and the first excited state can be determined by the (avoided) crossings between three linearly-perturbative energy levels (in s). The ground state (solid line in the figure) contains one avoided crossing, between the driver-Hamiltonian ground state (originating at s = 0 with energy E = −n) and the problem-Hamiltonian ground state (which originates at s = 1 with energy E = 0). The first excited state (dashed line) is characterized by two avoided crossings. The first one is between the problem-Hamiltonian ground state and the driver-Hamiltonian ground state, and a second one between the driver-Hamiltonian ground state and the first excited state of the problem Hamiltonian (originating at s = 1 with energy E = 1). The three states involved in the relevant avoided crossings are therefore: (i) The problem-Hamiltonian ground state which is a computational-basis state. (ii) The problem-Hamiltonian first excited state which is the equal superposition of n other computational-basis states, and (iii) The driver-Hamiltonian ground state which is the equal superposition of all computational-basis states. The avoided crossings between these three states can therefore be used to capture the gap and matrix element completely.

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Considering a reduced adiabatic evolution that is restricted to the subspace spanned by the above three states, the resultant three-level system can therefore be analytically analyzed. Specifically, it enables us to obtain the large-n asymptotic behavior of the gap:

$$\begin{equation} g(s)= \left\lbrace \begin{array}{@{}ccccc@{}}\displaystyle n-\frac{3 n -1}{2}s & \phantom{A} & 0 & <s \le & \displaystyle\frac{2n}{3n-1} \\ \displaystyle \frac{3 n -1}{2}s-n & \phantom{A} & \displaystyle\frac{2n}{3n-1} & <s \le & \displaystyle\frac{2n }{3n-3} \\ \displaystyle s & \phantom{A} & \displaystyle\frac{2 n}{3n-3} & < s \le & 1 \end{array}. \right. \end{equation} \tag{ 20 }$$

In the immediate vicinity of the minimum gap, we find that

$$\begin{equation} \min _s g(s)=\frac{2 n}{3} 2^{-n/2} , \end{equation} \tag{ 21 }$$

where the critical point approaches s_* = 2/3 for large n.

While the matrix element V₀₁(s) may also be computed using perturbation theory, it is easier to bound it from above by simpler considerations:

$$\begin{equation} V_{01}(s) =| \langle 0 | \mathrm{d}H / \mathrm{d}s | 1\rangle | = \frac{1}{s} | \langle 0 | \hat{H}_d |1\rangle | = \frac{1}{1-s}| \langle 0 | \hat{H}_p |1\rangle | . \end{equation} \tag{ 22 }$$

The above matrix elements for both the problem and driver Hamiltonian are easily bounded by their matrix norms to give:

$$\begin{equation} V_{01}(s) \le \min _s \left[\frac{n}{s},\frac{n}{1-s}\right] =2 n . \end{equation} \tag{ 23 }$$

While the above bound is by no means tight (the actual tight bound is also linear in n albeit with a smaller constant), the above bound will suffice for our purpose.

A constant-rate adiabatic scheme, following (2), will yield QAA running times that are O(max _sV₀₁(s)[min _sg(s)²]⁻¹) = O(2ⁿ/n), and would therefore provide no real advantage over the performance of classical algorithms.

Nonetheless, our ability to calculate the gap of the system accurately, along with proper bounds on the matrix element, allows for the construction of a locally-adiabatic path that would in turn yield significantly shorter QAA running times. The bound on the matrix element, combined with the gap g(s) found in (20) and (21) and the local adiabatic evolution condition (3), allows us to construct the locally-adiabatic path which, upon integration, yields the following scaling of the running time in the large n limit:

$$\begin{equation} \mathcal {T} = \int _0^1 \mathrm{d}s \left( \frac{\mathrm{d}s}{\mathrm{d}t }\right)^{-1} \sim O(\max _s V_{01}(s) [ \epsilon \min _s g(s)]^{-1}) =O\left(\frac{2^{n / 2}}{n \epsilon }\right) . \end{equation} \tag{ 24 }$$

To verify that the large-scale analysis presented above is correct, we plot in figure 5 the actual running time as it was calculated by integrating the exact-numerical gaps and matrix elements for up to n = 60 (crosses, the pre-factor has been removed) as a function of the number of input bits n, on a logarithmic scale. The solid line in the figure corresponds to $\epsilon \mathcal {T}=\frac{3}{2 n} 2^{n/2-1}$ which turns out to be the true large-n behavior of the running time, and which provides a verification for the large n scaling analysis presented above.

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We have thus shown how to construct a QAA with which the random energy model can be solved quadratically faster than any existing classical algorithm.

We note here that the above analysis is in accord with the bounds on the minimum complexity of the algorithm calculated in [8] by Farhi et al, which have shown that the algorithmic complexity of the QAA with either a one-dimensional projection as the driver Hamiltonian or an unstructured problem Hamiltonian (under certain conditions) is bounded from below by O(2^n/2) running times. Here, we have used both to explicitly construct an algorithm that achieves these bounds.

We have demonstrated how one can construct analogue, continuous-time, quantum algorithms for a family of unstructured problems, namely, 'scrambled input'. We have shown that for this type of problems, a compact set of dynamical equations may be written down explicitly, and subsequently solved or analyzed to enable finding optimized adiabatic or non-adiabatic paths, that yield algorithms that are more efficient than their corresponding classical algorithms.

We considered two specific examples. In the first, a simple prescription for deterministically solving the Deutsch–Josza problem was given. We have shown that the running time of the proposed analogue algorithm does not scale with the size of the problem, i.e., that it is exponentially faster than the classical algorithm and on par with the circuit-based quantum algorithm result.

In a second example, we provided an adiabatic solution to the problem of finding the ground state of a random energy model, in which the eigenvalues are taken from a binomial distribution but are then shuffled around. In this example, the provided algorithm was found to be, as one might expect, quadratically faster than the corresponding classical one.

It would be of interest to see how this family of problems may be further generalized, thereby expanding the scope of applicability of continuous-time quantum computing. We hope that the method presented here will open a way for other adiabatic as well as deterministic non-adiabatic faster-than-classical algorithms that would further demonstrate the power and potential encompassed in analogue continuous-time quantum computation. Specifically, we hope that the above algorithm will help in further pinpointing the precise equivalence between circuit-based and adiabatic quantum computing.

We thank Peter Young, Eleanor Rieffel and David Gosset for useful comments and discussions.