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Currently, it is not preciselly known whether quantum annealers bring any significant speed up. Lets take some task having exponential complexity on classical computer. If you run it on quantum annealer it will probably run faster. However the reason is not reduced complexity (it is still exponential) but smaller constants in function decribing the task ...


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This cannot yet be tested for more than about 100 qubits, since no publicly announced circuit-based quantum computer has been built with more than 127 qubits. Furthermore, the word "efficiency" needs to be more precisely defined: Are we talking about speed, energy efficiency, number of total qubits needed, or something else? I'm glad that in your ...


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For a problem of size $N$, computational complexity of quantum annealing is $\mathcal{O}(e^{\sqrt{N}})$. This is better than simulated annealing, which has a complexity of $\mathcal{O}(e^{N})$. Both these complexities are mentioned in this paper. More analysis needs to be done to answer this question quantitatively. But, qualitatively, we can expect to have ...


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The vertices and edges are the number of qubits and connections in DWaves quantum processing unit topology. It is called Chimera Graph, see DWaves docs . The graph is not fully connected. Therefore an embedding of the qubo problem has to be done. It is done heuristically, see minorminer docs. As an example, if your problem is fully connected, the maximum ...


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One way to do it would be to use a transformation, such as this one: \begin{align} X_i &= \frac{1 - Z_{i,j}Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{1}\\ Y_i &= \textrm{i}\frac{Z_{i,k}-Z_{i,j}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{2}\\ Z_i &= \frac{Z_{i,j}+Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{3}\\ I_i &= \frac{1 + Z_{i,j}Z_{i,k}}{...


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Consider $Q=H$. Let $|\lambda_n\rangle$ be the eigenbasis of $H$, i.e. $$ H=\sum_n\lambda_n|\lambda_n\rangle\langle\lambda_n|. $$ Now consider the $|x\rangle$ that minimises $\langle x|H|x\rangle$. Since the eigenbasis is an orthonormal basis, we can choose to write $$ |x\rangle=\sum_n\alpha_n|\lambda_n\rangle,\qquad\sum_n|\alpha_n|^2=1, $$ where it is now ...


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There are exactly solvable models, including for the most general target problem, that describes the performance without approximations. They show that the time of achieving the desired ground state is generally the same as in some classical algorithms but for special problems even an exponential speedup is possible. See, e.g., in https://arxiv.org/abs/2110....


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Problem solving in Ocean framework goes by modeling the problem into a BinaryQuadraticModel. The pure quantum samplers natively only understand this model. For the hybrid samplers (Leap) one can model the problem into DiscreteQuadraticModel. The bias are the linear terms in those models objective functions. They are derived with respect to the problem after ...


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Direct conversion There is a way to convert a QuadraticProgram(QP) from Qiskit into a BinaryQuadraticModel(BQM). First the QP has to be created with Qiskit. It can have linear constraints, integer variables and binary variables. The objective can have linear and quadratic terms. Quadratic constraints and float variables are not supported in the following ...


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The idea is that you want to use the adiabatic theorem. This states (roughly) that if your Hamiltonian $H(s)$ has an energy gap $\Delta(s)$ between the ground and excited states, then provided $$ \frac{d\Delta}{ds}\ll\epsilon, $$ for some small $\epsilon>0$ (if you look up the formal statements, there's a few more conditions, but this will do for this ...


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I am not familiar with the application you are trying to implement but see a general misunderstanding in the setting of the terms h_i and J_i,j . The numbers in the top of the output are the names of the variables you have defined. You are defining the values of the variables apple, ibm, microsoft, google as variable names. The setting of the biases h_i and ...


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The article of Devoret and Schoelkopf [1] and an update provided in Section 7.1 of Reagor [2] makes a comparison between Moore's law and an observed trend of exponentially improving $T_1$ and $T_2$ times for superconducting qubits. The trend they present shows a roughly exponential improvement from $10^0$ to $10^6$ nanoseconds for $T_2$ between various ...


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Yes for QUBO problems, vertices of the graph are qubits, and edges of the graphs are "couplers" which couple two qubits together. For example for the QUBO problem: $$\tag{1} b_1b_2 - 3b_1b_3 + b_3, $$ you need a machine with 3 qubits and 2 couplers (one between qubits 1 and 2, and one between qubits 1 and 3).


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