bath. Let us consider a thermal nonequilibrium process in a finite-time schedule t$_{0}$ = 0 ≤ t ≤ t$_{n}$ = n δ t . Ther- mal fluctuations can be simulated by the master equa- tion. The probability that the system is in a state σ$_{k}$ at time t$_{k}$ is denoted as P ( σ$_{k}$ ; t$_{k}$ ). The transition probabil- ity per unit time δ t is defined as M ( σ$_{k}$$_{+}$$_{1}$ | σ$_{k}$ ; t$_{k}$ ). In the original formulation of JE, the work is defined as the energy di ff erence merely attributed to the change of the Hamiltonian, but we can construct JE also in the case of changing the inverse temperature by defining the work as − β W ( t$_{k}$ ) = − ( β ( t$_{k}$$_{+}$$_{1}$ ) − β ( t$_{k}$ )) H$_{0}$ for the state σ . The left-hand side of JE can then be expressed as linear combination, similarly to the ordinary QA. If we change T → 0 very slowly, one can obtain the ground state of H$_{q}$ , which expresses the very low-temperature equilibrium state for H$_{0}$ , the cost function of the opti- mization problem that we wish solve. We however con- sider to construct a protocol with the same spirit as JE by using the special quantum system to overcome the bottleneck of the ordinary QA as proposed in the fol- lowing section. where ˜ P ( σ$_{0}$ ; t$_{0}$ ) denotes the initial equilibrium distribu- tion. Even if the transition term exp( δ tM ( σ$_{k}$$_{+}$$_{1}$ | σ$_{k}$ ; t$_{k}$ )) is removed in this equation, JE is trivially satisfied as one can simply confirm. A non-trivial feature of JE is in the insertion of the transition term. From Eq. (3), it is straightforward to prove JE by use of the detailed- balance condition. An observant reader may think the above formulation without any consideration of quan- tum nature is not available for the application to QA. Nevertheless we can apply the classical JE to QA by aid of the classical-quantum mapping [10]. The classical-quantum mapping leads us to a special quantum system, in which the (instantaneous) equilib- rium state of the above stochastic dynamics can be ex- pressed as the ground state. A general form of such a special quantum Hamiltonian is given as H$_{q}$ ( σ $^{′}$| σ ; t ) = I − e β ( t ) 2 $^{H$_{0}$}$M ( σ $^{′}$| σ ; t )e − β ( t ) 2 $^{H$_{0}$}$. This Hamiltonian has the ground state as | Ψ$_{eq}$ ( t $^{)}$〉 = ∑ σ e − β ( t ) H$_{0}$ / $^{2}$| σ 〉 / √$_{Z}$$_{(}$$_{t}$$_{). The}$ ground state energy is 0, which can be explicitly shown by the detailed-balance condition. On the other hand, the excited states have positive-definite eigenval- ues, which can be confirmed by the application of the Perron-Frobenius theorem. In the above special quantum system, we can deal with a quasi-equilibrium stochastic process as an adi- abatic quantum-mechanical dynamics in QA. Let us consider QA for the above special quantum system by setting the parameter corresponding to the temperature T → ∞ ( β → 0). This condition gives the trivial ground state for H$_{1}$ in the preceding section with an uniform This is essentially of the same form as Eq. (3). We measure the obtained state by the projection onto a specified state σ ′ . The probability is then given by |〈 σ $^{′}$| Ψ ( t$_{n}$ $^{)}$〉 | $^{2}$, which means that the ground state we wish to find is obtained with the probability proportional to $^{exp(}$− β ( t$_{n}$ ) H$_{0}$ ). If we continue the above procedure up to β ( t$_{n}$ ) ≫ 1, we can e ffi ciently obtain the ground state of H$_{0}$ . This is called the quantum Jarzynski anneal- ing (QJA) in the present study. Most of the readers, who are familiar with the ordinary computation, have considered that it may seem unnecessary to apply the time-evolution operator U ( σ$_{k}$$_{+}$$_{1}$ | σ$_{k}$ ; t$_{k}$ ), which expresses change between states by quantum fluctuations, at the middle step between the operations of the exponenti- ated work operators W$_{exp}$ ( t$_{k}$ ). The time-evolution op- erator does not mean an artificial control but describes the change by quantum nature during quantum com- putation, which is inherent property in quantum com- putation. However we here remember the nontrivial 2 We prepare a trivial ground state with a uni- form linear combination as the initial condition in the ordinary QA. This initial state corresponds to the high-temperature equilibrium state | Ψ$_{eq}$ ( t$_{0}$ $^{)}$〉 ∝ $^{exp(}$− β ( t$_{0}$ ) H$_{0}$ / $^{2)}$| σ 〉 with β ( t$_{0}$ ) ≪ 1. We introduce the exponentiated work operator W$_{exp}$ ( t$_{k}$ ) = $^{exp(}$−( β ( t$_{k}$$_{+}$$_{1}$ $^{)}$− β ( t$_{k}$ )) H$_{0}$ / 2). It looks like a non-unitary operator, but we can construct this operation by considering an extended quantum system [11, 12]. If we apply W$_{exp}$ ( t$_{k}$ ) to the preceding quantum state | Ψ$_{eq}$ ( t$_{k}$ $^{)}$〉, it is changed into a state corresponding to the equilibrium distribution with the inverse temperature β ( t$_{k}$$_{+}$$_{1}$ ). Af- ter then, the time-evolution operator U ( σ $^{′}$| σ ; t$_{k}$$_{+}$$_{1}$ ) = $^{exp(}$−i δ tH$_{q}$ ( σ $^{′}$| σ ; t$_{k}$$_{+}$$_{1}$ ) / $^{ℏ}$) also does not alter this state, since it is the ground state of H$_{q}$ ( σ $^{′}$| σ ; t$_{k}$$_{+}$$_{1}$ ). The result- ing state after the repetition of the above procedure is 4. Quantum Jarzynski annealing and its application 3. Classical-quantum mapping 〈 e − β W 〉 = ∑ { σ$_{k}$ } $^{n}$− 1 ∏ k = 0 { e − β W ( t$_{k}$ $^{)}$e δ tM ( σ$_{k}$$_{+}$$_{1}$ | σ$_{k}$ ; t$_{k}$ ) } × ˜ P ( σ$_{0}$ ; t$_{0}$ ) , (3) | Ψ ( t$_{n}$ $^{)}$〉 ∝ $^{n}$− 1 ∏ k = 0 { W$_{exp}$ ( t$_{k}$ ) U$_{k}$$_{+}$$_{1}$ ( σ$_{k}$$_{+}$$_{1}$ | σ$_{k}$ ; t$_{k}$ ) } . ×| Ψ$_{eq}$ ( t$_{0}$ $^{)}$〉 (4)