Method for quantum computing

Abstract

The disclosure presents a method for performing quantum computation The method comprises formulating a quantum Hamiltonian as a function of a first set of quantum states and providing a quantum simulator system, a set of quantum control systems, and a set of quantum resource systems. The method also comprises preparing the first set of quantum states on the set of quantum resource systems and applying a set of SWAP gates on the set of quantum simulator systems, the set of quantum control systems, and the set of quantum resource systems. The method also comprises determining the evolution of the set of quantum simulator systems by measuring one or more output quantum states from the set of quantum control systems or quantum resource systems.

Description

[0001]METHOD FOR QUANTUM COMPUTING FIELD OF THE DISCLOSURE This disclosure relates to quantum computing and more particularly to quantum simulation. BACKGROUND OF THE DISCLOSURE Quantum simulation is a promising and rapidly developing field which uses quantum features to solve difficult problems in science and technology. Simulation of the dynamics of quantum systems is an important part of quantum simulation theory. Hamiltonian simulation, which deals with simulation of unitary evolutions generated by a given Hamiltonian, has been vastly studied in the literature. Almost all of the proposed quantum simulation methods are gate-based, i.e., a sequence of quantum gates are applied to simulate the given Hamiltonian dynamics. The sequence of the gates in the gate-based quantum simulation are chosen based on the so called product formula. A quantum gate is an arrangement which performs unitary operations on a quantum system, such as a qubit. A quantum gate manipulates quantum states to perform computations. A quantum gate includes control procedures which are executed on the quantum system through control devices that are physically coupled to the quantum system. In other words, the control devices needed for performing these procedures on a given quantum system depend on the quantum technology which is being used. A gate- based simulation performed on a quantum computer may comprise a great number of unitary quantum gates that perform long sequences of operations on quantum systems. However, simulation of open-system dynamics is challenging for gate-based quantum computation where a sequence of unitary quantum gates are applied, because open quantum systems do not evolve unitarily. A general open system evolution is not even linear, and the generator of the dynamics depends on the state of the system and its correlation with the environment. The traditional quantum jump technique for simulation of the open system dynamics is applicable only to the linear evolutions. Some other gate-based simulation method has also been proposed recently, which are again applicable only to the linear case. There are also other physical situations, which lead to nonlinear effective Hamiltonians for the system dynamics. Bose-Einstein condensate and its Gross-Pitaveskii Hamiltonian which leads to a nonlinear Schrödinger equation is an example of such systems. Although there are other methods for quantum simulation such as linear combination of unitary evolutions, still they are based on application of quantum gates. Even in almost all of the recent (quantum) machine learning techniques for quantum simulation again applying a sequence of quantum gates forms the quantum neural network. But such neural networks have limited utility for machine learning purposes because they lack nonlinearity on the input data---the very source of power of neural network models. In other words, despite the power and usefulness of gate-based quantum computers, they have limitations. Specifically, quantum mechanics is a linear theory, but nonlinear models are prevalent and appear in aerodynamics, hydrodynamics, chemistry, engineering, artificial intelligence, and machine learning. Handling such modelling with gate-based quantum computers can be challenging. BRIEF DESCRIPTION OF THE DISCLOSURE An object of the present disclosure is to provide a method and computer program which overcome the above problem. The object of the disclosure is achieved by a method and computer program which are characterized by what is stated in the independent claims. The preferred embodiments of the disclosure are disclosed in the dependent claims. The disclosure is based on the idea of state-based quantum simulation. BRIEF DESCRIPTION OF THE DRAWINGS In the following, the disclosure will be described in greater detail by means of preferred embodiments with reference to the accompanying drawings, in which Figure 1 illustrates arrangements which may be used to perform the method. Figure 2 illustrates an SBQS algorithm. Figure 3 illustrates one embodiment of the method. DETAILED DESCRIPTION OF THE DISCLOSURE Here we introduce a new method for quantum simulation which is fundamentally different from the traditional gate-based method. Rather than applying a sequence of quantum gates, our method is based on preparing quantum states, hence we call it state-based quantum simulation (SBQS). The quantum states for SBQS can be chosen based on the ability of state preparations in the lab. For example, coherent quantum states can be chosen as the appropriate quantum states specially in the continuous case or the ``polarization states'', or any other chosen set based on which the Hamiltonian can be decomposed. Decomposition of the Hamiltonian in terms of quantum states changes the focus of simulation from interactions and gate preparation to state preparation. This provides a big opportunity for simulating various types of interactions without the need for their real existence. We then use density matrix exponentiation technique and show that only by applying controlled-SWAP (CSWAP) gates on prepared quantum states and some post processing / post selection, it is possible to simulate the given unitary evolution. Interestingly, this method is very powerful and versatile that makes it also suitable for simulation of open quantum system dynamics. Another feature of SBQS is that the instantaneous state of the evolving system itself can also be part of the state-based algorithm. Hence the current state of the system can also take part in its future evolution. This is the very key feature that gives the SBQS the ability for simulating nonlinear evolutions without the need for linearization. Linearization is the most common practice to solve nonlinear problems which is computationally costly and may even reduce the underlying features of the real model of interest. Just for the sake of analogy, the contrast between our SBQS and the GBQS can be seen better if one thinks of the contrast between the measurement-based quantum computation (MBQC) and the circuit-based (or gate-based) quantum computation (CBQC). In measurement-based quantum computation a highly enough entangled state is prepared and quantum algorithms are performed by applying measurements and single quantum gates, and the benefit is that there is no need to entangling quantum gates. MBQC has been shown to have equivalent computational power with CBQC. Here in the SBQS, however, we still need to use controlled-SWAP gate, which is an entangling universal gate. But the benefit is that we do not need to stick to unitary or even linear quantum evolutions. This makes our method also suitable for simulation of the open quantum system dynamics. We also believe that the proposed state-based and the traditional GBQS methods are not equivalent. We provide evidence from simulation of nonlinear Hamiltonians using the state-based method, which is exponentially more efficient than the traditional GBQS. At least, we cannot imagine any GBQS method that is equivalent to our suggested SBQS method in simulation of state-history dependent Hamiltonians. Thus, we believe that the state-based quantum simulation supersedes the gate-based quantum simulation at least in this specific problem. In the following we first introduce SBQS for simulation of the usual problems in the GBQS, i.e., simulation of standard Hamiltonians which are independent of the state of the system. We next generalize the formalism and show that SBQS can also be used for simulation of nonlinear Hamiltonians which are specific functions of the state of the system at the current or past times. As an application, we show that the solution of a set of nonlinear delayed differential equations can be obtained using SBQS of a nonlinear Hamiltonian. We also show how to simulate Gross-Pitaevskii Hamiltonian as an example of nonlinear Hamiltonian systems. We then show that how one can use the SBQS to simulate open quantum system evolutions. This disclosure describes a method for performing quantum computation. The method comprises formulating a quantum Hamiltonian as a function of a first set of quantum states. The method also comprises providing a quantum simulator system, a set of quantum control systems, and a set of quantum resource systems. The method also comprises preparing the first set of quantum states on the set of quantum resource systems. The method also comprises applying a set of SWAP gates on the set of quantum simulator systems, the set of quantum control systems, and the set of quantum resource systems. The method also comprises determining the evolution of the set of quantum simulator systems by measuring one or more output quantum states from the set of quantum control systems or the set of quantum resource systems. The method may, for example, be used to perform calculations which model the behavior of a physical system. In other words, the method may be used for simulation purposes. The simulation may, but does not necessarily need to, comprise time-dependence. In this disclosure, the term “quantum simulation” may refer to a simulation which is executed in a quantum computer, or to a simulation which is executed in a classical computer which simulates the quantum-mechanical elements of a quantum computer and their quantum behaviors. This disclosure also describes a computer system which is configured to perform the method. The system comprises means for performing the method. The method described in this disclosure may be executed with any kind of quantum computer hardware. It may, for example, be executed with a quantum processing unit which comprises (1) a superconducting system which utilizes two-level systems formed by Josephson junctions and capacitors, (2) a system which utilize trapped ions controlled by laser light, (3) a quantum dot system or diamond-nitrogen-vacancy systems where electron spin or charge states are manipulated with electromagnetic pulses, (4) a photonic system which utilizes linear or nonlinear optical devices and laser pulses or other light sources to implement operations, or (5) any other quantum computing system. In other words, the general principles described in this disclosure can be implemented with any quantum computing hardware setup. The method described in this disclosure may also be called a quantum algorithm. A computer system may comprise a classical processing unit and a quantum processing unit. The classical processing unit may be configured to provide control signals to the quantum processing unit such that the quantum processing unit performs the method. The quantum processing unit comprises means for performing the method. Alternatively, a computer system may comprise a classical processing unit which is configured to simulate the behavior of the quantum simulator system, the set of quantum control systems, the set of quantum resource systems, and the set of SWAP gates, and to perform the method by simulation. A quantum processing unit may be configured to perform the method. A computer program product may comprise instructions which, when executed by a computer, cause the computer to perform the method. A computer-readable medium may comprise instructions which, when executed by a computer, cause the computer to perform the method. The term “quantum state” may refer in this disclosure to two interrelated things: first, it may refer to the information about a quantum system which is presented mathematically in a density matrix and represented by the variable ρ. This information determines the possible behavior of the quantum system, such as how it might respond to measurements. Mathematical calculations may be performed with and on this abstract representation of a quantum state. Second, the term “quantum state” may refer to the actual physical state of a quantum system. Such quantum states may be measured. In this disclosure, formulating a quantum Hamiltonian as a function of a first set of quantum states may comprise a non-recursive procedure where each quantum state in the first set is represented by a predetermined density matrix, and a known Hamiltonian is decomposed as a function of these density matrices. Each quantum state in the first set may in this case be a state which can be easily prepared with the available experimental resources. Formulating the quantum Hamiltonian may alternatively comprise a recursive procedure where the density matrices of the first set of quantum states is obtained from earlier measurements performed on a quantum system in the quantum computer. This quantum system may be the quantum simulator system. In other words, earlier (physical) states of the quantum simulator system may be included in the formulation of the quantum Hamiltonian. Throughout this disclosure, the word “earlier” may refer to an earlier point in time in a time- dependent simulation. However, the method described in this disclosure may also be used for simulating events that lack time-dependence. Consequently, an alternative meaning of “earlier” (which may encompass the time-dependent meaning) may to a simulation step which preceded the current step. If the simulation is performed by executing the method in repeated program loops (1, 2, 3,.., N-2, N-1, and N) then the word “earlier” may, in loop N, refer to any data gathered in one of the loops N-1, N-2, etc, which preceded the current loop. The word “preceding” may be used in this disclosure with the same meaning as the word “earlier”. The method may also comprise providing a set of quantum simulator systems, a set of quantum control systems, and a set of quantum resource systems. In this disclosure, the term “providing” may refer to the preparation of quantum systems on a real world physical system. In other words, preparing these quantum systems may, for example, comprise building a qubit, a qudit, a quantum optical system, or any comparable quantum system in any of the hardware configurations listed above. Each set may comprise just one quantum system or multiple quantum systems. The quantum simulator system is a central component in this method because the result of the simulation is determined by its evolution. The quantum control systems may be used both for performing actions on the quantum simulator systems and for readout measurements. The resource quantum systems are used for carrying quantum states. As explained in more detail below, resource quantum systems may be the object of a measurement. The quantum simulator system and the resource quantum system may be labelled target systems. In other words, the set of target systems may comprise the quantum simulator systems alone, quantum resource systems alone, or quantum simulator systems and quantum resource systems together. The method also comprises the first set of quantum states in the set of quantum resource systems. In other words, the quantum systems which form the quantum resource systems are set into given states. These are the states on which the quantum Hamiltonian has a functional dependence. Actions which guide the evolution of the set of quantum simulator systems may be performed by applying one or more SWAP gates on the set of quantum simulator systems, the set of quantum control systems, and the set of quantum resource systems which have been arranged in the first set of quantum states. With the help of such SWAP gates, the states with which the Hamiltonian has been formulated can be utilized as quantum- mechanical operators which act on the quantum simulator system. The end result of the method may be the same as if the quantum Hamiltonian dynamics had been applied directly to the quantum simulator system, for example, in the form of a sequence of gates. However, the fact that the actions are in practice carried out only by SWAP gates and a set of quantum states opens up completely new possibilities in the design of quantum algorithms, as described in more detail below. Finally, after the SWAP gates have been applied, the evolution of the set of quantum simulator systems may be determined by measuring one or more output quantum states from the set of quantum control systems or the set of resource systems. Optionally, all or some of the resource systems or control systems may be discarded. The method may be repeated to extend the simulation for further time or steps. As mentioned earlier, the quantum Hamiltonian may be formulated as a function of a first set of quantum states. The function may, for example, be a weighted sum of the first set of quantum states. Each term in the weighted sum may comprise the product of a scalar real- or complex-valued coefficient and a quantum state from the first set of quantum states. This functional relationship is expressed in Equation 1 below. Other functional relationships may also be used. SBQS for Hamiltonian simulation We assume that a known Hamiltonian H is given, and we are going to simulate the unitary dynamics generated by H. This that H is known is an important factor in the standard GBQS. Here to explain the SBQS method we first start with the same assumptions and conditions that is usually assumed in the GBQS. Later, in the next sections, we show that how SBQS can be used for more general cases, e.g., when H is a function of the instantaneous and / or the past states of the system and hence is unknown. The SBQS method for known Hamiltonian simulation consists of three steps: (i) Decomposition of H in terms of quantum states: A set of density operators is chosen arbitrarily, provided that H can be expanded in terms of them. where coefficients and can be complex numbers in general. The set of density matrices, which is the first set of quantum states, can be chosen based on the lab ability for preparing quantum states. In other words, the first set of quantum states, through which the quantum Hamiltonian is formulated, can in principle be selected freely. However, this selection may be guided by practical considerations, and it may be optimized through experimentation. The scalar coefficients may in this case be independent of the earlier states of the quantum simulator system. It should also be noted that it does not matter whether the set of density operators is complete and if they are independent operators or not. However, just to give a general prescription for decomposition of any arbitrary H in terms of density operators, we point out two complete basis of density operators. (a) Discrete case: Decomposition of H in terms of polarization states:} We start with representation of H in the computational basis (or number-state basis), i.e., H = . Using the polarization identity states It is known that any operator can terms of coherent quantum states according to Sudarshan-Glauber P representation. Thus, for any H we have ii) Up to here, we showed that any Hamiltonian can be decomposed in terms of quantum states, i.e., a form like equation (1). Hence the evolution generated by a given H can be written as . We use Trotter-Suzuki expansion to break down U to single-density matrix, such that in which δ = t / n. evolutions governed by each term in the above equation. iii) Density matrix exponentiation} To simulate the evolution on a quantum system with initial state we do the following steps. First, quantum system with state is Next the exponential-SWAP gate is applied on the joint system's state . Here S is the SWAP gate and δ is a time step. The result would be in which we is the standard operator norm. Using n copies of process n times, where ht = nδ it is straightforward to show that To reach a of copies of should be As an alternative for gate UCS provided that an additional control system in the state is prepared: where the system is , otherwise nothing changes. In any embodiment presented in this disclosure, each or any SWAP gate in the set of SWAP gates may be a suitably adapted exponential-SWAP (ESWAP) gate, or any other quantum gate or combination of quantum gates to that effect. In any embodiment presented in this disclosure, each or any SWAP gate in the set of SWAP gates may be a controlled-SWAP (CSWAP) gate, or any other quantum gate or combination of quantum gates to that effect. In other words, all SWAP gates used in this method may, for example, be controlled-SWAP gates. Or some of the SWAP gates used in this method may, for example, be controlled-SWAP gates, or any other quantum gate or combination of quantum gates to that effect, and others may be exponential-SWAP gates, or any other quantum gate or combination of quantum gates to that effect. SBQS for designing dynamical systems Up to here we have assumed that a known Hamiltonian is given and we aimed to simulate the dynamics generated by the given Hamiltonian using the tools in SBQS, i.e., quantum states and ESWAP gates. We showed that the SBQS method can be used to simulate any given Hamiltonian. Here, however, we are going to study the reverse problem, i.e., assuming that we can use arbitrary quantum states and ESWAP gates what sort of Hamiltonians we can design using the SBQS? This question can be considered as a special form of a wider question: What can we achieve if we have access to quantum simulators? Can we get something bigger than what we see in standard quantum mechanics? With having access to classical simulation, if we have enough classical resources, we can simulate quantum dynamics which are beyond the classical systems. Now assuming that we have access to quantum simulation, if we have enough quantum resources, can we achieve something bigger than standard quantum physics? This is a wider and more inclusive question that we cannot answer completely. However, we attack this question using our SBQS formalism and show that within the SBQS framework we can achieve dynamical systems which are beyond those that are expected within the standard quantum mechanics. The key point here, is that the SBQS method allows us to use the state of the system itself as one of the elements in Hamiltonian design or more generally in designing the quantum algorithms. This, as we show later, leads to possibility of designing nonlinear quantum systems, either closed or open, which are not allowed within the framework of the standard quantum mechanics in which the (closed system) evolution is linear. Although it has been already shown in the literature that, in some rare cases, nonlinear effective theories can explain the behavior of some specific quantum systems, such systems are still considered as exceptional and weird. Here, we show using more than one copies of the system and the SBQS formalism, which consists of valid elements of standard quantum mechanics, we can achieve a wide variety of nonlinear quantum dynamical systems, i.e., going beyond the standard quantum physics. There is a nonlinear element in standard quantum mechanics, i.e., measurement, but we have never been able to generate a nonlinear closed system, i.e., a nonlinear Hamiltonian, using measurements. However, the SBQS method provides this possibility that we can design a nonlinear Hamiltonian with or without using the measurements. In the next section we explicitly show how to design a specific given family of nonlinear Hamiltonians. We then show how this can be used to solve a class of nonlinear differential equations. Figure 1 illustrates a schematic figure of an SBQS algorithm according to any embodiment presented in this disclosure. The circled system 11 is the system of interest whose dynamics is going to be controlled and is initially in the state . This is the quantum simulator system. Other systems 33, which are quantum systems, may have state and may be copies of the simulator system at an initial time. The rest of the quantum resource systems 33 are ancillary systems and can initially be in any arbitrary state as given in Equation 1. The systems 21, 22, and 23 are the set of quantum control system. They may, for example, be used as control qubits. Each control qubit is connected with a dashed-line to an edge. The corresponding edge indicates the target systems on which the SWAP gate is applied. The circle 12 around a state means a measurement is performed on that system. Each segment t1, t2, t3 shows one time step on the evolution of the system of interest. In this schematic figure three time steps of simulation is performed on the system of interest. For each time step the corresponding procedure may be highlighted by different colors and tagged with a time label may be performed. Existence of several copies of the simulator system in this procedure leads to nonlinear evolution of the system of interest. More generally, figure 1 illustrates the quantum simulator system 11 in the middle. The scheme illustrated in this figure may be applied in any embodiment presented in this disclosure. It can be seen that three different branches (illustrated here as simulation steps t1, t2 and t3) of interconnected quantum control systems and quantum resource systems extend outward from the quantum simulator system 11. Each branch may represent a particular arrangement of quantum computing hardware around the quantum simulator system, which is used to perform one or more simulation steps. In practice, the number of branches may be freely selected. The interconnections between the various systems may in some cases be rearranged during the simulation and between simulation steps. Based on the state of a quantum control system (21, 22, 23), a SWAP gate (41, 42, 43, 44) performs an action on the adjacent quantum resource systems (31, 32, 33, 34), and in some cases the quantum simulator system 11, which are connected to the adjacent quantum control system as figure 1 illustrates. This action is a swapping action where the states of the systems are flipped. The dynamics of the quantum simulator system 11 are determined by these actions. The SWAP gate 41, for example may act on the three connected systems (11, 21, 31) by exchanging the states of the two target systems (11 and 31) depending on the state of the control system (21). In other words, if the system 21 was in state A, before the SWAP gate was applied, and systems 11 and 31 were in states B and C, respectively, then the state of system 21 does not change after the SWAP gate was applied and the state of system 11 may, for example, be C and the state of system 21 may be B after the SWAP gate was applied---depending on what A is. As figure 1 illustrates, some of the quantum resource systems 31, 32, 33, 34 may, at some point in time during the simulation process, be in the same state as the quantum simulator system 11. The quantum resource systems 31, 32, 33, alternatively be in a predetermined state ρiwhich does not necessarily correspond to the state of the quantum simulator system. Alternatively or additionally, in the embodiment which is described in more detail below, some of the quantum resource systems 31, 32, 33, 34 may, at some point during the simulation process, be in the state where quantum simulator system 11 was at an earlier point in time. The quantum Hamiltonian determines which states should be prepared in the quantum resource systems before the method is performed. In each branch of figure 1, the set of quantum control systems comprises a first quantum control system 21, and the set of quantum resource systems comprises a first quantum resource system 31. The set of SWAP gates comprises a first SWAP gate (schematically illustrated by the dashed line 41) which is configured to act on the quantum simulator system 11, the first quantum control system 21, and the first quantum resource system 31. The one or more output quantum states may, for example, be measured (12) from the first quantum control system 21. However, other options are also possible. The one or more output quantum states could alternatively be measured from any other quantum control system 22-23, or from one of the quantum resource systems 31-34. It is illustrated in figure 1 that the state may exist also in one or more quantum resource systems 31-34. In some cases, the state which is carried by a quantum resource system 31-34 may be an earlier state of the quantum simulator system 11. The state which is being carried by a particular quantum system 11, 21-23, 31-34 at any given simulation step depends on the detailed implementation of the quantum computing arrangement. In branch t1, the set of quantum control systems also comprises a second quantum control system 22, and the set of quantum resource systems also comprises a second quantum resource system 32, and the set of SWAP gates comprises a second SWAP gate (schematically illustrated by the dashed line 42) which is configured to act on the second quantum control system 22 and the first (31) and second (32) quantum resource systems. In branch t1 the set of quantum resource systems also comprises two third quantum resource systems 33, and the set of SWAP gates comprises a third SWAP gate (schematically illustrated by the dashed line 43). The third SWAP gate is configured to act on the second quantum control system 22 and the third quantum resource systems 33. In branch t2 the set of quantum resource systems also comprises two third quantum resource systems 33, and the set of SWAP gates comprises a third SWAP gate (schematically illustrated by the dashed line 43), and the third SWAP gate is configured to act on the first quantum control system 21 and the third quantum resource systems 33. In branch t2 the set of quantum control systems also comprises additional quantum control systems 23, and the set of quantum resource systems also comprises additional quantum resource systems 34 connected in series with one of the two third quantum resource systems 33. The set of SWAP gates comprises additional SWAP gates (schematically illustrated by the dashed lines 44), and each additional SWAP gate is configured to act on a pair of quantum resource systems in the series. Extension to nonlinear (state-history dependent) Hamiltonians Here we provide a SBQS algorithm for designing a specific family of nonlinear Hamiltonians. The class of Hamiltonians that we consider here is quite general and can be considered as a nonlinear generalization of equation 1 where the couplings, can depend on the state of the system rather than being constant, i.e., where ci is a constant real number and σi is a given quantum state. Here where are the states of the system in the past times and nj is a 0 leads to dependence on the instantaneous state of the system). Figure 2 illustrates an SBQS algorithm for simulating nonlinear Hamiltonian. In this figure one time step of the dynamics governed by one of the sentences in the Hamiltonian of equation 12. It is important to note that, here, the Hamiltonian is not known because the states of the system are unknown. Only the prescription of the Hamiltonian in terms of the state (at previous / current times) t – ai is given. Therefore, the usual GBQS methods cannot be used for simulation of such a dynamical system. To simulate this Hamiltonian using the SBQS method we first use Trotter expansion which means that we only need to simulate Hamiltonians like In other words, each by an earlier state of the quantum simulator system. Another important point is that, the system evolving with this Hamiltonian is a memory system, i.e., its evolution in a time step depends on the state of the system at previous times. For solving the Schrödinger equation of such systems, or for simulating their history- dependent dynamics, it is not sufficient to have only the initial state of the system, rather we need to know all the states in the initial time interval, from t=0 to τ = maxi{ai}, where τ is the memory length of the system. With this information, we do the following steps for simulating the evolution of the system which is in the state to the next time step τ+δ where (see fig.2): (a) Prepare a control system in quantum state which is normalized up as a control system for application of a CSWAP gate. (b) Prepare the target systems in the state , in which is an N-partite system, where , and for each i there are ni separate systems in the state . Hence target systems that we count them starting from 0. (ii) Apply a concatenation of N separate CSWAP gates on adjacent target systems (each SWAP gate is applied on the corresponding target state) and (simulator systems) when the control system is in the state ) and then trace out the target systems to find the updated control system, where is the CSWAP gate on the updated control system and the new target systems in the state : (iv) Apply a and keep the state only when the result of the measurement is . Then trace out the control system and the first target system which leads to the following state of the system: (v) To go further states. This is because, as we mentioned earlier, the system is a memory system. Hence when needs be we have to repeat the simulation up to the time that a copy of the system in the needed past time is created, so that we can use this copy to simulate the evolution in the future times. This is the cost that we need to pay to be able to simulate a nonlinear system using the linear standard quantum mechanics theory. In other words, the first set of quantum states may comprise quantum states which correspond to the state of the quantum simulator system at one or more preceding points in time. Furthermore, the states of the quantum control systems in the set of quantum control systems may be partly determined by the state of the quantum simulator system at one or more preceding points in time. The flowchart in figure 3 illustrates one example of how the method may be carried out in practice. In simulation loop j, the preparation (61) of the control qubits in the quantum control systems corresponds to Equation 13. Resource states are also prepared (63) on the quantum resource systems. These resource states σjmay comprise states which are equal to the state of the quantum simulator system at an earlier point in time. The preparation (62) of target systems corresponds to Equation 14. A C-SWAP gate is then applied (64) between these systems. After that, the resource and simulator states (65) may be discarded, and the quantum control systems are in an updated control state (Equation 15 above). The method is then performed in Equation 16 by utilizing the quantum control systems (51), preparing new resource states (53), and inputting the earlier state of the quantum simulator (the end result of loop j-1) 52 into the C-SWAP gate 54. The output quantum state 56 of the quantum simulator system in this simulation step j may then be determined according to Equation 17 by measuring the state of the control qubits and discarding the resource states (55). The states 53 may also include earlier states of the quantum simulator from other loops, such as j-1, j-2, j-3, and so on. The method introduced above under the heading SBQS for Hamiltonian simulation may be implemented using only the lower half of the flowchart in figure 3 (steps 51-56). The control qubits are in this case simply prepared in predetermined states which do not depend on the earlier states of the quantum simulator system. Alternatively, as described above, the upper half of the flowchart may be utilized for preparing particular states 66 in the quantum control systems. Application: SBQS of nonlinear delay differential equations Assume a set of nonlinear delay differential equations (NLD) are given as: where time t and fmn is a variables at N different times, i.e., , in which li is a natural number showing the highest degree of instant t-ai, where a0 = 0. Thus, any fmn can be written generally as in which at each the degree of each variable xi at the given time. The first D elements are the degrees of D variables at time t – a0, the second D variables are the degrees of D variables at time t – a1, and so on. Each rijis an integer between 0 and . To show how SBQS technique can be used to solve the above equation, we map this problem into the simulation of a nonlinear dynamical system introduced in the previous section (to simplify the notation we omit time-dependence of variables where there is no ambiguity). To do so we need to follow several steps: (i) We add two extra equations to the set in equation 18. One equation is added for a new independent fixed parameter x0: which means that f00 = 1 and fm0 = f0n = 0, forall m and n. This constant parameter added for some technicalities which becomes clear later. Another equation is added for a normalizing variable such that about , we r is chosen such that the similar to other fmn s becomes a polynomial of all variables like the one in equation 19 in which the index i runs from 0 to D+1. It should be noted that, if needs be, we first scale all variables so that the normalization becomes possible. Now we define the normalized quantum state: It is be retrieved using a Schrödinger-like equation in which is a nonlinear this dynamical evolution can be considered beyond standard quantum mechanics, however, the SBQS method which is well within the framework of the standard quantum mechanics can successfully simulate this dynamics. (ii) To transform HNLDinto the form of equation 1 with nonlinear couplings, it is sufficient to write fmn in the form of equation 11. To do this we note that if rij is even, which is the expectation value of operator with to the of the state of the system at the pertinent time, i.e., . If rij is odd, using the constant parameter x0we can write in terms value of an which is the respect to the state . For simplicity and generality, we assume in the even. Accordingly, we can rewrite any fmn given in equation 19 as the expectation value of an operator with respect to such that identity introduced below equation 7, we can write where Tr1indicates collective SWAP that can be obtained by multiplication of relevant number of local SWAPs. Using this, it is straightforward to see that all terms in HNLDcan be written in the desired form of equation 12 and hence simulatable using the SBQS technique. As an example, we brought details of the SBQS for solving logistic map differential equation, , in the SM, where we expanded the inverse of normalization variable x2, up to O((1 – x2)2). Thus, we need 3 copies of the state of the system at each time to be able to simulate every single term of the Trotter-Suzuki expansion and the number of the terms in the Trotter expansion is equal to 35. It should be noted that the number of Trotter terms completely depends on the decomposition of the Hamiltonian in terms of quantum states and in our case it has been done just with a brute-force approach and as a proof of principle for showing the possibility of the SBQS method. SBQS of open quantum system dynamics Up to here it has become clear that the SBQS applications are not restricted to the unitary evolutions. Here we show that how an open quantum system dynamics can be simulated with the SBQS method. The mostly used form for the evolution of an open quantum system is given by a Lindblad dynamical equation in which the Lindbladian is given by Here we do not put any assumption on the sign of γi so the evolution can be either Markovian or non-Markovian. To show that the SBQS technique can be applied also on the open system case we proceed as follows: (i) Assuming that a known Lindblad superoperator is given we use a vectorization technique to transform it to an equation similar to the Schrödinger equation: An operator A represented as in the computational basis is transformed to a vector , where is a vector larger Hilbert space--- this definition and the identity (with T being transposition), we can obtain the vectorized Lindblad basis such that where is the and is the counterpart of in the vectorized space. We use this vectorized form of the evolution to implement SBQS simulation. (ii) We prepare a quantum system at the initial state where is a density matrix, not necessarily a normalized quantum state. The normalization in equation 27 should be considered as a scaling of the variables of equation 26, , while the equation remains unchanged with the scaled variables. (iii) is similar to the Schrödinger equation with a non-Hermitian Hamiltonian. Similar to what we did previously for simulation of such evolutions, we first expand in terms of quantum states in which s are the space. This equation is similar to equation 1 for which we have already developed our simulation protocol. Following the steps of simulation of linear Hamiltonian dynamics, we obtain However, the is instead the normalized quantum state (iv) Now we should read the information of the physical state from the simulated state . To revert this state back into the density matrix form original system, we note . Hence , from which the system density reconstructed as It is simple to see that the case of nonlinear open system dynamics, similar to what we did for the nonlinear Hamiltonians. Hence SBQS open system simulation covers a large family of open system evolutions. The SBQS provides a novel and powerful quantum simulation method which is substantially more efficient and intrinsic to a broader class of problems compared to the widely used gate-based method. In fact, the very nonlinear nature of SBQS makes it conducive to solving and simulating nonlinear models and problems. Nonlinear models appear everywhere from weather forecast, aerodynamics, hydrodynamics, pharmatech, etc. to nonlinear history-dependent Hamiltonians and open quantum systems. All these indicate that there is a vast range of immediate technological as well as scientific problems that can benefit from the luxury offered by SBQS. SBQS substantially enhances and customizes quantum simulation by designing the simulation protocol based on the ability of a lab to prepare states. In addition, the nature of the SBQS method is such that it can be used to solve nonlinear problems without the need for linearization. Linearization is the most common practice to solve nonlinear problems in the current gate-based method which is computationally costly and may even reduce the underlying features of the real model of interest. In SBQS it is possible to use the very state of the system itself as part of the algorithm. This means that the current state of system can take part in its future evolution. This is the very key feature that gives the SBQS the ability for simulating nonlinear evolutions. By obviating the need for linearization using SBQS, the computational cost drops drastically. The method described in this disclosure may for example be used for solving non-linear delayed differential equations, or for simulating open quantum systems. Furthermore, the method may also be used for preparing a state-based quantum gate.

Claims

CLAIMS 1. A method for performing quantum computation, characterized in that the method comprises: − formulating a quantum Hamiltonian as a function of a first set of quantum states, − providing a quantum simulator system, a set of quantum control systems, and a set of quantum resource systems, − preparing the first set of quantum states on the set of quantum resource systems, − applying a set of SWAP gates on the set of quantum simulator systems, the set of quantum control systems, and the set of quantum resource systems, − determining the evolution of the set of quantum simulator systems by measuring one or more output quantum states from the set of quantum control systems or the set of quantum resource systems.

2. A method according to any preceding claim, wherein the first set of quantum states comprises quantum states which correspond to an earlier state of the quantum simulator system.

3. A method according to claim 2, wherein the states of the quantum control systems in the set of quantum control systems are partly determined by an earlier state of the quantum simulator system.

4. A method according to any preceding claim, wherein the function is a weighted sum of the first set of quantum states, and each term in the weighted sum comprises the product of a scalar coefficient, real- or complex-valued, and a quantum state from the first set of quantum states.

5. A method according to claim 4, wherein each scalar coefficient is at least partly determined by an earlier state of the quantum simulator system.

6. A method according to claim 4, wherein all scalar coefficients are independent of earlier states of the quantum simulator system 7. A method according to any preceding claim, wherein the set of quantum control systems comprises a first quantum control system, and the set of quantum resourcesystems comprises a first quantum resource system, and the set of SWAP gates comprises a first SWAP gate which is configured to act on the quantum simulator system, the first quantum control system and the first quantum resource system, and the one or more output quantum states are measured from the first quantum control system.

8. A method according to claim 7, wherein the set of quantum control systems also comprises a second quantum control system, and the set of quantum resource systems also comprises a second quantum resource system, and the set of SWAP gates comprises a second SWAP gate which is configured to act on the second quantum control system and the first and second quantum resource systems.

9. A method according to claim 8, wherein the set of quantum resource systems also comprises two third quantum resource systems, and the set of SWAP gates comprises a third SWAP gate, and the third SWAP gate is configured to act on the second quantum control system and the third quantum resource systems.

10. A method according to claim 7, wherein the set of quantum resource systems also comprises two third quantum resource systems, and the set of SWAP gates comprises a third SWAP gate, and the third SWAP gate is configured to act on the first quantum control system and the third quantum resource systems.

11. A method according to claim 10, wherein the set of quantum control systems also comprises additional quantum control systems, the set of quantum resource systems also comprises additional quantum resource systems connected in series with one of the two third quantum resource systems, and the set of SWAP gates comprises additional SWAP gates, and each additional SWAP gate is configured to act on a pair of quantum resource systems in the series.

12. A method according to any preceding claim, wherein each SWAP gate in the set of SWAP gates is an exponential-SWAP gate.

13. A method according to any of claims 1-11, wherein each SWAP gate in the set of SWAP gates is a controlled-SWAP gate.

14. A computer system configured to perform the method of any of claims 1-13.

15. A computer system comprising a classical processing unit and a quantum processing unit, the classical processing unit being configured to provide control signals to thequantum processing unit such that the quantum processing unit performs the method of any of claims 1-13.

16. A computer system comprising a classical processing unit which is configured to simulate the behavior of the quantum simulator system, the set of quantum control systems, the set of quantum resource systems, and the set of SWAP gates, and to perform the method of any of claims 1-13 by simulation.

17. A quantum processing unit configured to perform the method of any of claims 1-13.

18. A computer program product comprising instructions which, when executed by a computer, cause the computer to perform the method of any of claims 1-13.

19. A computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the method of any of claims 1-13.

Images