JP2010054938A - Quantum simulator, quantum computer and method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To simulate spin-spin interaction. <P>SOLUTION: The quantum simulator includes: N pieces of physical systems having one-to-one correspondence to respective apexes of a finite undirected graph which has two lower states ¾0>, ¾1> and two upper states ¾2>, ¾3> and has N pieces of apexes and edges between the apexes; a plurality of optical systems 1000 which includes an optical system for coupling an optical resonator (j, j') including a physical system j and an optical resonator (j', j) including a physical system j' with an optical resonator as the optical system corresponding to an edge (j, j') and an optical system for coupling a transition between ¾1>1-¾2> of the physical system j with resonator mode of the optical resonator (j, j'), and has one-to-one correspondence to respective edges; an optical source 1101 for supplying a plurality of laser beams which are coupled with a transition between ¾1>-¾2>, a transition between ¾0>-¾3> and a transition between ¾1>-¾3> of a plurality of physical systems; a plurality of modulators 1104 for adjusting frequency, intensity and phase of the plurality of laser beams; and a controller 1105 for controlling the plurality of modulators. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a quantum simulator, a quantum computer, and a method using coupling between an optical resonator and a physical system.

  In recent years, research on “quantum simulation” has been actively conducted to realize a quantum many-body model that is difficult to solve strictly in a physical system with good controllability such as cooling atoms. Among them, quantum simulation of an interacting spin system model represented by the Ising model is particularly important. One of the reasons is that it is possible to observe the behavior of individual spins in an interacting spin system, and that it is possible to realize difficult things with ordinary magnetic spins. Quantum simulation of the Ising model is also important from a viewpoint other than system-specific problems.

  First, when an appropriate Ising model can be realized, a type of entangled state called a cluster state can be easily generated (see, for example, Non-Patent Document 1). It is known that if a cluster state is prepared, a quantum calculation in which each individual spin is regarded as a qubit can be executed using only one qubit gate and reading of the qubit (for example, see Non-Patent Document 1). . If the Ising model includes an interaction between a spin and a magnetic field in at least two directions, an arbitrary one qubit gate can be realized by the interaction, and such an Ising model quantum simulator becomes a quantum computer as it is.

Next, if quantum simulation of the Ising model is possible, there is a possibility that the problem of obtaining the lowest energy state (ground state) in the Ising model can be efficiently solved by a technique called adiabatic quantum computation (see, for example, Non-Patent Document 2). . This problem is closely related to the optimization problem (energy is a cost function) and is very important, but it is a so-called NP-hard problem that is difficult to solve with an ordinary computer.
R. Raussendorf and HJ Briegel, Phys. Rev. Lett. 86, 5188 (2001). M. Grajcar, A. Izmalkov, and E. Il'ichev, Phys. Rev. A 71, 144501 (2005). MJ Hartmann, FGSL Brandao, and MB Plenio, Phys. Rev. Lett. 99, 160501 (2007).

  As a proposal of a physical system that effectively realizes the Ising model, one based on so-called resonator quantum electrodynamics has been proposed (see, for example, Non-Patent Document 3). In the method of Non-Patent Document 3, a plurality of microresonators arranged corresponding to the spin arrangement of the Ising model to be realized and atoms coupled to these resonators are used. Spin is expressed using two stable states of an atom. The two resonators corresponding to the two interacting spins are directly coupled, and the atom-atom (spin-spin) is effectively utilized by utilizing the atom-resonator interaction and the resonator-resonator interaction. Realize interaction. Here, when a microresonator such as a microsphere resonator or a photonic crystal resonator is directly coupled via a near field, one atom is added to each resonator of such a coupled microresonator system. It is difficult to combine them at the current technical level. In addition, considering a Fabry-Perot resonator and the like, there is a method for realizing direct coupling between a resonator and a resonator (as used in Non-Patent Document 3) using a medium such as an optical fiber. Not self-evident. Therefore, when an Ising model quantum simulator based on resonator quantum electrodynamics is considered, for example, a proposal of a principle / configuration that is easier to implement than Non-Patent Document 3 is desired.

  The present invention has been made in view of the above circumstances, and an object thereof is to provide a quantum simulator, a quantum computer, and a method for simulating a spin-spin interaction.

  In order to solve the above problem, the quantum simulator of the present invention has two lower states | 0>, | 1> and two upper states | 2>, | 3> having higher energy than the lower states, A finite undirected graph having edges connecting N vertices and certain vertices (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j < j ′)) N physical systems one-to-one corresponding to each vertex (the physical system corresponding to vertex j is defined as physical system j), and an optical system corresponding to the side (j, j ′). , Including an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and an optical fiber that couples the optical resonator. A plurality of optical systems one-to-one corresponding to each side, including an optical system in which a transition between> − | 2> is coupled to a resonator mode of the optical resonator (j, j ′) A light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the physical system of It comprises a plurality of modulators that adjust the frequency, intensity, and phase of light, and a control device that controls the plurality of modulators.

  The quantum computer of the present invention has two lower states | 0>, | 1> and two upper states | 2>, | 3> having higher energy than the lower states, and there are N vertices between vertices. Each vertice of a finite undirected graph having an edge connecting vertices (the j-th vertex is a vertex j and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)) One-to-one corresponding N physical systems (the physical system corresponding to the vertex j is defined as the physical system j) and the optical system corresponding to the side (j, j ′), including the physical system j. An optical resonator (j ′, j) including a resonator (j, j ′) and a physical system j ′ and an optical fiber coupling the optical resonator, and a transition between | 1> − | 2> of the physical system j A plurality of optical systems one-to-one corresponding to each side, including an optical system coupled to the resonator mode of the optical resonator (j, j ′), and transition between | 1> − | 2> of the plurality of physical systems | 0>-| 3> transition, | 1>-| 3> light source for supplying a plurality of laser beams coupled with the transition, and a plurality of modulations for adjusting the frequency, intensity, and phase of the plurality of laser beams And a control device for controlling the plurality of modulators, and using the lower states | 0> and | 1> of the plurality of physical systems as qubits, the states of all the physical systems are set to (| 0 > + | 1>) / √2, and the plurality of modulators are controlled so that all the optical systems have control phase inversion gates for qubits of two physical systems included in them. And performing quantum computation using the cluster state obtained as a result.

The quantum computer of the present invention has two lower states | 0>, | 1> and two upper states | 2>, | 3> having higher energy than the lower states, and there are N vertices between vertices. Each vertice of a finite undirected graph having an edge connecting vertices (the j-th vertex is a vertex j, and the edge connecting vertices j and j ′ is an edge (j, j ′) (j <j ′)) One-to-one corresponding N physical systems (the physical system corresponding to the vertex j is defined as the physical system j) and the optical system corresponding to the side (j, j ′), including the physical system j. An optical resonator (j ′, j) including a resonator (j, j ′) and a physical system j ′ and an optical fiber coupling the optical resonator, and a transition between | 1> − | 2> of the physical system j A plurality of optical systems one-to-one corresponding to each side, including an optical system coupled to the resonator mode of the optical resonator (j, j ′), and | 1> − | 2> transition between the plurality of physical systems; | 0>- 3> inter-transition, | 1>-| 3> a light source for supplying a plurality of laser beams coupled to the inter-transition, a plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams, A control device for controlling the modulators of the plurality of systems, and using the lower states | 0>, | 1> of the plurality of physical systems as qubits, controlling the plurality of modulators, The initial state of the physical system is a Hamiltonian H ₀ where the ground state becomes the ground state, and the plurality of modulators are controlled so that the Hamiltonian of the entire system is changed from the Hamiltonian H ₀ to the Hamiltonian H for which the ground state is desired. The ground state of the Hamiltonian H is obtained by changing adiabatically so as to always remain in the ground state.

  According to the quantum simulator, quantum computer and method of the present invention, the spin-spin interaction can be simulated.

Hereinafter, a quantum simulator, a quantum computer, and a method according to embodiments of the present invention will be described in detail with reference to the drawings. Note that, in the following embodiments, the same numbered portions are assumed to perform the same operation, and repeated description is omitted.
The Ising model in this embodiment is a model of a physical system composed of a group of spins, and its Hamiltonian is composed of an interaction term between an external magnetic field and spin and an interaction term between spin and spin, and the external magnetic field and spin. The interaction term is expressed by the inner product of the magnetic field vector and the spin vector, and the spin-spin interaction term is expressed by multiplying the product of one directional component of two interacting spins by the coupling constant. The model is shown. An undirected graph G (V, E) consisting of vertices corresponding to spins on a one-to-one basis (S is a set of vertices of G, and E is a set of edges of G). Expressed by the sides of The model parameters are the three directional components of the external magnetic field and the coupling constant.

Next, the quantum simulator of the present invention will be described in detail with reference to FIGS.
It is assumed that the spin-spin interaction of the Ising model to be realized is expressed by an undirected graph G (V, E). V is a set of vertices of G, and the j-th vertex is called vertex j. The vertices have a one-to-one correspondence with spins, and if the number of spins is N, the number of vertices is also N. E is a set of sides of G, and the side connecting vertex j and vertex j ′ is called side (j, j ′) (j <j ′).

  FIG. 1 is an energy level diagram of a physical system used in the present invention. The stable lower states | 0> and | 1> are considered as the downward and upward states of the spin, respectively. There are N physical systems, one-to-one correspondence with the vertices. A physical system corresponding to the vertex j is called a physical system j.

Two optical resonators 201 and 202 and an optical fiber 203 that couples them to the physical system j 204 and the physical system j ′ 205 in a one-to-one correspondence with the side (j, j ′) of G (V, E). The optical system consisting of is installed as shown in FIG. An optical system corresponding to the side (j, j ′) is referred to as an optical system (j, j ′). In addition, two optical resonators constituting the optical system (j, j ′) are an optical resonator (j, j ′) 201, an optical resonator (j ′, j) 202, and an optical fiber (j , J ′) 203. Each of the optical resonator (j, j ′) 201 and the optical resonator (j ′, j) 202 has a resonator mode (j, j ′) 251 and a resonator mode (j ′, j) 252.
FIG. 3 shows an example in the case of three spins. This is a case where an Ising model in which three spins are arranged at the vertices of a triangle and all the spins interact with each other is simulated.
3 includes a resonator including the physical system 1 304, a resonator including the physical system 2 305, and a resonator including the physical system 3 306, and optical fibers 301, 302, and 303 that connect the resonators. The resonator including the physical system 1 304 and the resonator including the physical system 2 305 are connected by the optical fiber (1, 2) 301 and are respectively connected to the resonator mode (1, 2) 351 and the resonator mode (2, 1). 352. The resonator including the physical system 2 305 and the resonator including the physical system 3 306 are connected by the optical fiber (2, 3) 302 and are respectively connected to the resonator mode (2, 3) 353 and the resonator mode (3, 2). 354. The resonator including the physical system 3 306 and the resonator including the physical system 1 304 are connected by the optical fiber (1, 3) 303 and are respectively connected to the resonator mode (3, 1) 355 and the resonator mode (1, 3). 356.

The transition between | 1> − | 2> of the physical system j 204 is coupled to the resonator mode (j, j ′) 251 by the coupling constant g _{j, j ′} (see FIG. 1). Further, each physical system is irradiated with a laser coupled with the transition between | 1>-| 2>, | 0>-| 3>, and | 1>-| 3>, and the rabbi frequency is set to Ω. Let ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) , and detuning be Δ ₂ , Δ ₃ , and Δ ₃ , respectively. | 1> - | 2> detuning between the laser and the cavity mode of binding between the transition and delta _c.

  The optical resonator of FIG. 3 is a Fabry-Perot resonator in which a plane mirror and a spherical mirror face each other, and the resonator mode and the fiber mode are coupled via the plane mirror. (Such coupling between the resonator mode and the physical system has already been realized. For example, M. Trupke, J. Goldwin, B. Darquie, G. Dutier, S. Eriksson, J. Ashmore, and EA (See Hinds, Phys. Rev. Lett. 99, 063601 (2007).) All resonators are assumed to have the same resonance frequency. An optical fiber is also sandwiched between plane mirrors to form a kind of resonator, and its resonance frequency is assumed to be approximately equal to the resonance frequency of the resonator. 1 example).

  In the present invention, one resonator mode and one transition are coupled, whereas in the past (for example, see MJ Hartmann, FGSL Brandao, and MB Plenio, Phys. Rev. Lett. 99, 160501 (2007)). Two resonator modes and two transitions that share the excited state are coupled, and this also complicates the detuning setting in this example. Also in this respect, the present invention has an advantage that it is easier to realize than the above-described known examples.

The Hamiltonian of this system is generally represented by the following formula (1) (for example, see A. Serafini, S. Mancini, and S. Bose, Phys. Rev. Lett. 96, 010503 (2006)).

The following (2).

Here, a _{j, j ′} are the annihilation operator of the resonator mode (j, j ′), b _{j, j ′} are the annihilation operator of the optical fiber (j, j ′), and v is the optical resonator and the optical fiber. _{, J j, j ′} is the detuning between the fiber mode and the resonator mode, φ _j, j′≡2πl _{j, j ′} / λ _c (λ _c is the resonance of the resonator in the fiber) The wavelength, l _{j, j ′} is the length of the optical fiber (j, j ′)) is the phase shift due to the fiber, and σ _kl ^(j) ≡ | k> _jj <l |. The following (3) is the sum for all coupled spin pairs (j, j ′) (j <j ′), and the following (4) is the sum for all physical systems (j is the number of the physical system). Represents. Unlike the above known example, the resonator mode is not coupled directly but coupled via the fiber mode.

Here, assuming the following (5), (6), and (7), the time evolution of the system is approximately determined by the effective Hamiltonian of the following equation (8). In the present invention, it is important to use this effective Hamiltonian.

Here, the following (9) and (10) to (13).

Here, (14) below is the sum of vertices j ′ connected with the vertex j and the side (j, j ′), and (15) below is about the vertex j ′ connected with the vertex j and the side (j ′, j). Represents the sum of

Further, H _eff is the Hamiltonian of the Ising model to be realized, and it can be seen that quantum simulation of the Ising model can be executed by the quantum simulator of the present invention shown in FIGS. Model parameters J _{j, j ′} , h _x ^(j) , h _y ^(j) , h _z ^(j) of the Ising model are controlled by Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) . Yes, the rabbi frequency can be controlled by controlling the intensity and phase of the laser light to be emitted.

Detuning delta _j of the resonator mode and the fiber _{mode, j ',} the above inequality _{| v | »| Δ c +} Δ j, j' | be smaller to the extent satisfying.

In order to confirm the validity of the above approximation, a numerical simulation using a complete Hamiltonian H was performed in the case of FIG. 3 and compared with the case of calculation using H _eff . The starting state was set to the following (16) (the last | 0> represents the vacuum state for all resonator modes and fiber modes).

The comparison results are shown in FIGS. 4 shows the case (17) below, and FIG. 5 shows the case (18) below.

The vertical axis represents fidelity between the state | Ψ (t)> using H and the state | Ψ _eff (t)> using H _eff : F (t) = | <Ψ _eff (t) | Ψ (t)> | ² . Simulation time was up to _{t = π / J j, j} '= π / 3.13 × 10 -4 g. In both the case of FIG. 4 and FIG. 5, the fidelity is very close to 1, indicating that the above approximation is well established. In the calculation using H, the excitation of the physical system, the optical resonator mode, and the fiber mode is considered to be sufficiently small, and the physical system, the resonator mode, and the fiber mode are not excited at all. Was carried out using only the excited state. The fidelity very close to 1 also indicates the validity of this approximation. It can be seen that the present invention makes it easy to realize an Ising model quantum simulator based on resonator quantum electrodynamics.

  Next, a method for performing quantum computation using the quantum simulator of the present invention will be described. This method is based on a technique called one-way quantum computation (see, for example, R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001)).

  Consider the quantum simulator of the present invention that can simulate an Ising model of a square lattice 601 or a hexagonal lattice 602 as shown in FIG. It is known that the cluster state of the square lattice 601 or the hexagonal lattice 602 is universal in performing quantum computation (any quantum computation can be performed using this cluster state) (for example, R. Raussendorf and HJ Briegel, Rev. Lett. 86, 5188 (2001) and M. Van den Nest, W. Dur, A. Miyake, and HJ Briegel, New J. Phys. 9, 204 (2007)). For example, in the case of a hexagonal lattice 602, the actual configuration of the apparatus may be achieved by connecting a system in which one physical system 703 and three resonator modes 751 shown in FIG. Although the angle formed by the optical fiber is not 120 degrees, the optical fiber bends, and the graph does not matter because only the way of connection is a problem. The optical resonator is a Fabry-Perot resonator in which a plane mirror 702 and a spherical mirror 701 face each other.

First, the states of all physical systems are prepared in the following (19).

Next, each physical system is set so that the parameters of the Ising model are J _{j, j ′} = h _z ^(j) = J, h _x ^(j) = h _y ^(j) = 0 (J is arbitrary). Are irradiated with lasers of appropriate frequency (detuning Δ ₂ , Δ ₃ ) and intensity / phase (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ) for a time π / J Only develop time and stop laser irradiation. Thus, when spins are identified as qubits, the control phase inversion gate is executed for all interacting spin pairs, and the resulting physical system state is an entangled state called a cluster state. (See, for example, R. Raussendorf and HJ Briegel, Phys. Rev. Lett. 86, 5188 (2001)). Quantum calculation can be performed by sequentially reading one qubit gate and qubit from the end of the cluster state thus obtained. Note that the calculation in FIG. 4 is a simulation of triangular cluster state generation.

  Next, a method for obtaining the lowest energy state (ground state) of the Ising model using the quantum simulator of the present invention will be described. This problem is related to the optimization problem (energy is a cost function) and is an important problem.

Simulated annealing or quantum annealing is known as an algorithm for solving this problem with an ordinary computer. Quantum annealing is the following method. First, consider a simple Hamiltonian H ₀ , consider changing the Hamiltonian slowly over time, and finally setting the Hamiltonian H to know the ground state. For this time-varying Hamiltonian, the start state ground state of H ₀ | to simulate the time evolution at the time of the G>. Then, the ground state of H that the user wanted to know is obtained as the final state. This method is based on quantum adiabatic evolution (if the Hamiltonian time change is sufficiently slow, the state of the system that was initially in the ground state remains in the time-varying Hamiltonian ground state). In ordinary computers, there are 2 ^N equations to be solved to simulate N spins (Schrödinger equation), so it is generally difficult to solve and simulate the Schrödinger equation. An approximate solution like this is used.

  A quantum calculation method called adiabatic quantum calculation completely performs the simulation necessary for the above quantum annealing by using a quantum computer. If the quantum simulator of the present invention is used, the problem of obtaining the ground state of the Ising model can be solved by this adiabatic quantum calculation. Therefore, the quantum calculation method using the quantum simulator of the present invention can be applied to all problems to which quantum annealing based on the Ising model can be applied.

Consider the quantum simulator of the present invention that can simulate the corresponding Ising model according to the Hamiltonian form of the Ising model whose ground state is desired. As a specific example, consider the problem of obtaining a ground state when J _{j, j ′} = h _z ^(j) = h ₀ , h _x ^(j) = h _y ^(j) = 0 (h ₀ is a positive real number). .

First, the states of all physical systems are prepared in the above (19).
Then, the parameters of the Ising model ^{_{h x (j) = -h,}} J j, j '= h z (j) = h y (j) (h is a positive real number) so that, each physical The system is irradiated with a laser having an appropriate frequency (detuning Δ ₂ , Δ ₃ ) and intensity / phase (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ). At this time, h is gradually increased from 0. Then, since the state of the above equation (19) is always a Hamiltonian eigenstate, the state of the physical system remains unchanged from the above equation (19). When h reaches h ₀ , h _x ^(j) = − h ₀ (1-t / T), J _{j, j ′} = h _z ^(j) = h ₀ t / T, h _y ^(j) = 0 The intensity and phase of the irradiating laser (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ) are adjusted so that Here, at time t, t = 0 when the process is started. The calculation is complete when t = T.

(First embodiment)
Examples of the quantum simulator of the present invention will be described. The case of the three spins shown in FIG. 3 will be described, but the other cases can be similarly implemented.

As a physical system, Pr ³⁺ ions doped in Y ₂ SiO ₅ crystal are used. It is possible to use cooled atoms trapped in an optical cavity as is actually done (for example, J. McKeever, A. Boca, AD Boozer, JR Buck, and HJ Kimble, Nature 425, 268 (2003)), the use of impurities in the solid has the advantage that the coupling constant does not change with time because the position does not change. The above-mentioned Pr ^3+: As in Y ₂ SiO _5, rare earth ions doped into the crystal has the property of absorbing only light of the direction of polarization with respect to the crystal axis, Fabry - Perot resonator polarization related Another feature is that only one of the two modes needs to be considered. As shown in FIG. 8, two of the three hyperfine states of the ground state ³ H ₄ are used as stable lower states | 0> and | 1>, and among the three hyperfine states of the excited state ¹ D ₂ Are used as the upper states | 2> and | 3>. The remaining states are set to | 4> and | 5> as shown in FIG.

By mirror processing the crystal surface, three optical resonators are formed for one crystal 901 as shown in FIG. At this time, these resonator modes 751 are made to intersect at one point on the same plane, and one Pr ³⁺ ion at this intersection is used. It should be noted that by making the concentration sufficiently low, it is possible to have only one Pr ³⁺ ion at this position and having an energy level as shown in FIG. Those having a transition which is resonant with the cavity mode at other Pr ³⁺ ions resonator mode is negligible by transferring to the ground state without resonance the resonator mode by optical pumping.

Direction of the crystal axis, the polarization direction of the absorption of the Pr ³⁺ ions is maximized choose to be perpendicular to the plane spanned by the resonator mode 751. The resonator mode 751 is coupled to the optical fiber 704 via the plane mirror 702 as shown in FIG. 9, and three crystals 1004 to 1006 are connected by the optical fibers 1001 to 1003 as shown in FIG. Optical fibers other than the optical fibers connecting the crystals (reading optical fibers 1007, 1009, and 1011) are connected to photon detectors 1008, 1010, and 1012 to detect photons leaking from the reading resonators.

  The resonator modes 1051 to 1056 of the optical resonator coupled to the optical fiber couple with the | 1> − | 2> transition at an appropriate detuning. The resonator modes (readout resonator modes) 1057 to 1059 of the optical resonator not coupled to the optical fiber resonate with the | 4> − | 5> transition (see FIG. 8).

An overall view is shown in FIG. The optical system 1000 of FIG. 10 is maintained at the liquid helium temperature (4K) in the cryostat 1107 except for the photon detectors 1008, 1010, 1012. As the light source ³ H ₄ - ¹ D ₂ the ring dye laser 1101 that is frequency stabilization of wavelength 606nm of the transition it is used. The modulator 1104 includes an acousto-optic effect element (AOM) and an electro-optic effect element (EOM), and controls the frequency and intensity of the laser beam by AOM and the phase of the laser beam by EOM. Laser light from the light source is divided into a large number of beams by a beam splitter 1102, and three of them are irradiated to three readout resonators so as to resonate with the resonator mode from the spherical mirror side. The remaining laser beam is applied to the intersection of the resonator modes in the three crystals. Although only three are shown in FIG. 11, in reality, more laser beams are used as described above.

The parameter setting will be described.
As shown in FIG. 8, the detuning is set to Δ ₂ / 2π = −Δ ₃ / 2π = 200 kHz and Δ _c / 2π = 20 kHz.

The coupling constant v between the optical resonator and the optical fiber is set to v / 2π = 9 MHz by setting the resonator length l _c to 5 mm, the fiber length l _f to 5 cm, and the transmittance T of the plane mirror to 10 ⁻⁴ . v satisfies the following equation (20).

Here, c is the speed of light in vacuum, n _c is the refractive index of the Y ₂ SiO ₅ crystal, about 1.8, and n _f is the refractive index of the fiber core, about 1.5. Detuning delta _j between the fiber mode and the resonator _{modes, j 'is,} delta _{j, j'} to meet / 2π«27MHz, designing fiber length.

The coupling constants g _{j, j ′} and g ′ between the ion and the optical resonator are set to 60 kHz by selecting the radius of curvature of the spherical mirror. Since the coupling constant is inversely proportional to the root of the cross-sectional area of the resonator mode at the ion position, the radius of curvature of the spherical mirror is designed so that the cross-sectional area becomes an appropriate value.

The remaining parameters are Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , and Ω ₃₁ ^(j) , which control the intensity and phase of the laser light applied to the ions by AOM and EOM as described above. It is adjusted by that.

As described above, the case where the numerical calculation is performed in FIGS. 4 and 5 can be performed.
In this embodiment, Δ ₂ = −Δ ₃ = 20 g and Δ _c = 2g are established as in FIGS. 4 and 5, and the Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) Can be adjusted by controlling the intensity and phase of the laser light applied to the ions by AOM and EOM as described above, and φ _{j, j ′} = 0 by adjusting the phase of Ω ₂₁ ^(j) . Only v is v = 100 g in the case of FIGS. 4 and 5, whereas in this embodiment, v = 9.6 × 10 ² g, which is greatly different, but is used to derive the aforementioned H _eff . The approximation is valid if v is sufficiently large, and the obtained H _eff does not depend on v. Therefore, the present embodiment and the cases of FIGS. 4 and 5 coincide.

The spin (quantum bit) is measured as follows. First, the state | 0> is converted to the state | 4> by a laser pulse resonating with the | 0>-| 3> transition and the | 4>-| 3> transition (K. Bergmann, H. Theuer, and BW Shore Rev. Mod. Phys. 70, 1003 (1998)). Next, a weak laser beam resonating to the readout resonator is irradiated from the spherical mirror side, and the leakage is detected by the photon detector. If a photon is detected, the state of the qubit at the end of the calculation is | 1>, and if no photon is detected, the state of the qubit at the end of the calculation is | 0>. (This method is based on the phenomenon of vacuum Rabi splitting. Assume that the incident laser is weak enough to observe vacuum Rabi splitting.)
(Second embodiment)
An example of unidirectional quantum computation using the quantum simulator of the present invention will be described. The general theory of one-way quantum computation using the quantum simulator of the present invention has been described in detail in the above embodiment. A specific operation method for realizing this with the apparatus of the first embodiment will be described. Hereinafter, a method of generating a 3-qubit cluster state and a quantum calculation method using the same using the apparatus of the first embodiment will be described. The same can be done for other cluster states.

First, the state of all Pr ³⁺ ions is prepared in the state of (19). For this purpose, a ring dye laser is used to first irradiate each ion with a laser beam resonating with the | 1>-| 3> transition and | 4>-| 5> transition for a while and initialize it to | 0> by optical pumping. Next, an optical pulse resonating with the | 0>-| 3> transition and the | 1>-| 3> transition (which can be converted into a pulse waveform by the AOM) is irradiated to each ion, and the state of (19) is obtained by an adiabatic passage. (See, for example, K. Bergmann, H. Theuer, and BW Shore, Rev. Mod. Phys. 70, 1003 (1998)).

Next, each Pr ³⁺ ion is set so that the parameters of the Ising model are J _{j, j ′} = h _z ^(j) = J, h _x ^(j) = h _y ^(j) = 0 (J is arbitrary). Laser light set to an appropriate intensity and phase (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ) by AOM and EOM is irradiated from a ring dye laser, and only during time π / J Develop time and stop laser irradiation. Thus, a cluster state is obtained.

  If a cluster state is obtained, the quantum calculation can be performed using a qubit read and a 1-qubit gate. The reading method has been described in the first embodiment. One qubit gate can be implemented by irradiating a light pulse resonating with | 0>-| 3> transition, | 1>-| 3> transition and | 4>-| 3> transition from a ring dye laser (H Goto and K. Ichimura, Phys. Rev. A 75, 033404 (2007)).

(Third embodiment)
An embodiment of the adiabatic quantum computation using the quantum simulator of the present invention will be described. The general theory of the adiabatic quantum computation using the quantum simulator of the present invention has been described in detail in the above embodiment. A specific operation method for realizing this with the apparatus of the first embodiment will be described. Hereinafter, the adiabatic quantum calculation for obtaining the ground state of the triangular Ising model using the apparatus of the first embodiment will be described. The other cases can be similarly implemented.

As a specific example, when J _{j, j ′} = h _z ^(j) = h ₀ , h _x ^(j) = h _y ^(j) = 0 (h ₀ is a positive real number) described in detail in the above embodiment. Consider the problem of finding the ground state.
First, the states of all physical systems are prepared in the state (19). This can be performed in the same manner as in the second embodiment.

Next, each Pr is set such that the parameters of the Ising model are h _x ^(j) = − h, J _{j, j ′} = h _z ^(j) = h _y ^(j) (h is a positive real number). ^{The 3+} ions are irradiated from the ring dye laser 1101 with laser light set to an appropriate intensity and phase (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ) by AOM and EOM. At this time, h is gradually increased from 0. When h reaches h ₀ , h _x ^(j) = − h ₀ (1-t / T), J _{j, j ′} = h _z ^(j) = h ₀ t / T, h _y ^(j) = 0 Thus, the intensity and phase (Rabi frequencies Ω ₂₁ ^(j) , Ω ₃₀ ^(j) , Ω ₃₁ ^(j) ) of the laser beam irradiated by the modulator 1104 (AOM and EOM) are adjusted. Here, at time t, t = 0 when the process is started. The calculation is complete when t = T.

  The result can be read out by the reading method described in the first embodiment.

  According to the embodiment described above, the spin-spin interaction is simulated using the interaction between the Fabry-Perot optical resonator and the physical system and the resonator-resonator interaction via the optical fiber. A quantum simulator, a quantum computer and a method can be provided.

  Note that the present invention is not limited to the above-described embodiment as it is, and can be embodied by modifying the constituent elements without departing from the scope of the invention in the implementation stage. In addition, various inventions can be formed by appropriately combining a plurality of components disclosed in the embodiment. For example, some components may be deleted from all the components shown in the embodiment. Furthermore, constituent elements over different embodiments may be appropriately combined.

The figure which shows the mode of the coupling | bonding of the energy level of a physical system used for the quantum simulator of this invention, and each transition and a laser beam and a resonator mode. The figure which shows the optical resonator and optical fiber which are used in order to simulate a spin-spin interaction. The figure which shows the quantum simulator of this invention when simulating the Ising model which arrange | positions three spins at the vertex of a triangle and all spins interact. The figure which shows a numerical simulation result. The figure which shows a numerical simulation result. The figure which shows the undirected graph showing the spin-spin interaction in the case of performing a quantum calculation using the quantum simulator of this invention. The figure which shows the basic element of the apparatus for simulating the Ising model of the hexagonal lattice of FIG. In ^Examples, Pr _3+: Y 2 SiO ₅ energy levels of ^{Pr 3+} ions in the crystal, and a diagram showing a state of coupling between each transition and the laser-cavity modes. FIG. 4 is a diagram illustrating resonator-processed Pr ³⁺ : Y ₂ SiO ₅ crystals and optical fibers coupled to the resonators utilized in the examples. The figure showing the main optical systems in an Example. 1 is an overall view of a quantum computer of an embodiment.

Explanation of symbols

201, 202 ... optical resonator, 203, 301-303, 704, 1001-1003 ... optical fiber, 204, 205, 304-306, 703 ... physical system, 251, 252, 351-356, 751 , 1051 to 1056 ... resonator mode, 601 ... square grating, 602 ... hexagonal grating, 701 ... spherical mirror, 702 ... plane mirror, 901, 1004 to 1006 ... crystal, 1000 ... Optical system, 1007, 1009, 1011, 1057 to 1059 ... Reading optical fiber, 1008, 1010, 1012 ... Photon detector, 1101 ... Ring dye laser, 1102 ... Beam splitter, 1104 ... Modulator, 1107 ... Cryostat.

Claims (13)

It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one A physical system (a physical system corresponding to the vertex j is a physical system j);
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Corresponding multiple optical systems,
A light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
A plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
And a control device for controlling the plurality of modulators.   The quantum simulator according to claim 1, wherein the physical system is a rare earth ion doped in a crystal.   The physical system includes a lower state | 4> and an upper state | 5> in addition to two lower states | 0> and | 1> and two upper states | 2> and | 3>, and is included in the optical system. 3. The quantum simulator according to claim 1, further comprising a plurality of optical resonators that resonate with a transition between | 4> − | 5> apart from the optical resonators.   The quantum simulator according to claim 3, further comprising a measuring device that measures a state of the physical system using a plurality of optical resonators that resonate with the transition between | 4> − | 5>. It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one A physical system (a physical system corresponding to the vertex j is a physical system j);
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Corresponding multiple optical systems,
A light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
A plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
A controller for controlling the plurality of modulators,
Using the lower states | 0>, | 1> of the plurality of physical systems as qubits and setting all the physical systems to (| 0> + | 1>) / √2, the plurality of modulators Control, perform a control phase inversion gate for the qubits of the two physical systems included in all the optical systems,
A quantum computer that performs quantum computation using the cluster state obtained as a result. It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one A physical system (a physical system corresponding to the vertex j is a physical system j);
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Corresponding multiple optical systems,
A light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
A plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
A controller for controlling the plurality of modulators,
The lower states | 0>, | 1> of the plurality of physical systems are used as qubits, and the plurality of modulators are controlled so that the Hamiltonian of the entire system becomes the Hamiltonian H ₀ whose initial state of the physical system is the ground state. age,
By controlling the plurality of modulators, the Hamiltonian of the entire system is adiabatically changed from the Hamiltonian H ₀ to the Hamiltonian H that wants to know the ground state so that the state of the system always remains in the ground state. A quantum computer for obtaining a ground state of the Hamiltonian H.   The physical system includes a lower state | 4> and an upper state | 5> in addition to two lower states | 0> and | 1> and two upper states | 2> and | 3>, and is included in the optical system. The quantum computer according to claim 5, further comprising a plurality of optical resonators that resonate with a transition between | 4> − | 5> apart from the optical resonators to be provided.   The quantum computer according to claim 7, wherein the state of the physical system is measured using a plurality of optical resonators that resonate with the transition between | 4> − | 5>. The Hamiltonian consists of an interaction term between an external magnetic field and spin and an interaction term between spin and spin, and the interaction term between the external magnetic field and spin is expressed by the inner product of the magnetic field vector and the spin vector. Is an quantum simulation method for simulating an Ising model expressed by multiplying the product of one directional component of two interacting spins with a coupling constant,
It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one Prepare a physical system (the physical system corresponding to the vertex j is the physical system j),
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Prepare corresponding optical systems,
Preparing a light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
Preparing a plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
The side corresponds to the spin-spin interaction, the Hamiltonian of the N physical systems corresponds to the Hamiltonian of the Ising model, and the downward state and the upward state of the spin of the Ising model are respectively represented by the down state | 0> and | 1>
Coupling between the resonator mode and the transition between | 1>-| 2> of the physical system, coupling between the resonator mode and the fiber mode of the optical fiber, and between | 1>-| 2> of the physical system By utilizing the coupling between the transition, | 0>-| 3> transition, | 1>-| 3> transition and laser light, the interaction between the spin and the spin and the interaction between the external magnetic field and the spin are obtained. Simulate,
The frequency, intensity, and phase of a plurality of laser beams supplied from the light source by the plurality of modulators are controlled to adjust the three directional components of the external magnetic field, which are model parameters of the Ising model, and the coupling constant. Quantum simulation method characterized by The physical system includes a lower state | 4> and an upper state | 5> in addition to two lower states | 0> and | 1> and two upper states | 2> and | 3>, and is included in the optical system. In addition to the optical resonator, a plurality of optical resonators that resonate with the transition | 4> − | 5> are further prepared,
The quantum simulation method according to claim 9, wherein the state of the physical system is measured using a plurality of optical resonators that resonate with the transition | 4> − | 5>. It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one Prepare a physical system (the physical system corresponding to the vertex j is the physical system j),
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Prepare corresponding optical systems,
Preparing a light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
Preparing a plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
Preparing a control device for controlling the plurality of modulators;
Using the lower states | 0>, | 1> of the plurality of physical systems as qubits and setting all the physical systems to (| 0> + | 1>) / √2, the plurality of modulators Control, perform a control phase inversion gate for the qubits of the two physical systems included in all the optical systems,
A quantum computation method characterized in that quantum computation is performed using a cluster state obtained as a result. It has two lower states | 0>, | 1> and two upper states | 2>, | 3> that have higher energy than the lower state, and a finite number of edges that connect N vertices to certain vertices. An undirected graph (the j-th vertex is a vertex j, and the edge connecting the vertex j and the vertex j ′ is an edge (j, j ′) (j <j ′)), N pieces corresponding one-to-one Prepare physical system (physical system corresponding to vertex j is physical system j),
An optical system corresponding to the side (j, j ′), an optical resonator (j, j ′) including the physical system j, an optical resonator (j ′, j) including the physical system j ′, and the light Including an optical fiber that couples the resonator, and an optical system in which the transition between | 1>-| 2> of the physical system j couples to the resonator mode of the optical resonator (j, j ′). Prepare corresponding optical systems,
Preparing a light source for supplying a plurality of laser beams coupled to | 1>-| 2> transition, | 0>-| 3> transition, | 1>-| 3> transition of the plurality of physical systems;
Preparing a plurality of modulators for adjusting the frequency, intensity, and phase of the plurality of laser beams;
Preparing a control device for controlling the plurality of modulators;
The lower states | 0>, | 1> of the plurality of physical systems are used as qubits, and the plurality of modulators are controlled so that the Hamiltonian of the entire system becomes the Hamiltonian H ₀ whose initial state of the physical system is the ground state. age,
By controlling the plurality of modulators, the Hamiltonian of the entire system is adiabatically changed from the Hamiltonian H ₀ to the Hamiltonian H that wants to know the ground state so that the state of the system always remains in the ground state. A quantum calculation method characterized by obtaining a ground state of the Hamiltonian H. The physical system includes a lower state | 4> and an upper state | 5> in addition to two lower states | 0> and | 1> and two upper states | 2> and | 3>, and is included in the optical system. In addition to the optical resonator, a plurality of optical resonators that resonate with the transition | 4> − | 5> are further prepared,
The quantum computation method according to claim 11, wherein the state of the physical system is measured using a plurality of optical resonators that resonate with the transition between | 4> − | 5>.

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