Methods for obtaining solutions to multi-product formulas

Abstract

Examples are disclosed involving obtaining solutions to m-order multi-product formulas to solve quantum computing problems involving product formulas. One example provides a method comprising selecting a set of exponents k j , where each k j is a real number and is an exponent of a linear combination in the form of a product formula. Based on the set of exponents k j , a set of pre-factors a j The determination is based on an underdetermined solution of an mxM linear system of equations, where M is the number of low-order product formulas in the linear combination of the product formulas. The set of indices k j and the set of pre-factors a j is used to solve the quantum computing problem including the product formula. By minimizing the set of exponents k j and the set of pre-factors a j , a sparse solution of the multi-product formula is generated, thereby reducing computation time and scaling.

Description

Background Art

[0001] Simulations of chemical systems can be used to reduce trial-and-error losses by predicting the properties of chemical substances and the expected outcomes of reactions. Since quantum computers are based on quantum mechanics, quantum simulations can be used to simulate the behavior of electrons, atoms, and / or molecules in a system more efficiently than classical computing systems. Instead of consuming computational resources exponentially (e.g., 2 for an n-electron system), n ), a quantum computer may only use a polynomial number of resources. This allows simulations of much larger systems without relying on approximations that reduce the accuracy of the resulting predictions. Summary of the Invention

[0002] This summary is provided to introduce a selection of concepts that are further described below in the detailed description in a simplified form. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that solve any or all disadvantages noted in any part of this disclosure.

[0003] In one disclosed example, a method for obtaining a solution to an m-order multi-product formula to solve a quantum computing problem including a product formula is provided. One example provides a method comprising selecting a set of exponents k j , where each k j is a real number and is the exponent of a linear combination of product formulas. Based on a set of exponents k j , a set of pre-factors a j Determined based on underdetermined solutions of an m×M system of linear equations, where M is the number of low-order product formulas in the linear combination of product formulas. A set of exponents k j and a set of prefactors a j It is used to solve quantum computing problems involving product formulas. By minimizing a set of exponential k j and a set of prefactors a j , sparse solutions of the multi-product formula are generated, thus reducing computation time and scaling. BRIEF DESCRIPTION OF THE DRAWINGS

[0004] Figure 1 An example method for obtaining solutions to m-order multi-product formulas to solve quantum computing problems involving product formulas is shown.

[0005] Figure 2 shows that using the classical product formula or Figure 1 Variations of the method plot multiple queries against the integrator order.

[0006] Figure 3 Various aspects of an example quantum computer are schematically shown.

[0007] Figure 4 Illustration of a Bloch sphere, which graphically represents the quantum state of a quantum computer's qubit.

[0008] Figure 5 An example classical computing device is schematically shown. DETAILED DESCRIPTION

[0009] The curse of dimensionality underlies the difficulty of simulating quantum systems. Because the dimensionality of a Hilbert space grows exponentially with the number of particles, a classical computer requires exponential time to accurately reproduce quantum dynamics. Fortunately, quantum computers show great promise in solving this problem.

[0010] Many physical systems of interest (such as in superconductivity, chemistry, and general quantum field theory) are characterized by Hamiltonian operators. They are the sum of a polynomial number of N local terms. Even though each term modifies the amplitude of the wave function exponentially, a quantum computer can still apply the time evolution by having a polynomial number of physical resources.

[0011] A series of quantum gates can be generated that approximate the time evolution of a Hamiltonian operator. This time evolution operator can then be called as a subroutine in an additional quantum algorithm. For example, quantum algorithms can be used to factor large numbers, which is important in many cryptographic applications.

[0012] Just as there are many algorithms for multiplying numbers or matrices together, there are many ways to design quantum circuits that implement this time-evolution operator. All of these different algorithms have different trade-offs, unique strengths, and unique weaknesses. Some of them are tailored to the particular structure of the problem being simulated, while others are tailored to different types of problems or structures.

[0013] There is usually a trade-off between the cost and complexity of implementing an algorithm. For example, to multiply two matrices of dimension nxn, the complexity of the traditional method is n 3 The complexity of the best possible sequence of operations is n 2.3 However, this algorithm is somewhat impractical due to the presence of large constant factors applied to n. There is considerable interest in designing quantum systems that operate more efficiently on quantum computers. Improvements come from reducing the exponent and decreasing the value of the constant factor. In most of these cases, the performance of an implementation can be quantified in terms of scaling relative to the simulation time, number of terms, and target error, as well as whether the exponents of these terms are smaller.

[0014] Early quantum simulation algorithms were based on an idea called Trotterization. To simulate a Hamiltonian operator as a sum of many terms, the time evolution of the Hamiltonian operator can be approximated by applying the time evolution to each term in the sequence. This quantum simulation can be performed efficiently on a quantum computer, for example, in polynomial time and cost with respect to the number of terms and target error.

[0015] However, Hamiltonians and other physical systems are described by more parameters than just time and the number of terms. For example, given a system with two molecules far away from each other, it can be assumed that these two molecules do not interact with each other. Therefore, the cost of simulating these two pseudo-independent molecules should be only twice the cost of simulating one molecule. The cost of simulating n molecules should scale linearly with n. However, the most general quantum simulation algorithms actually scale quadratically with respect to the number of terms n, because the distance structure between terms is not introduced. For the Hamiltonian, these algorithms assume the worst case, such as that every term can interact with every other term. Therefore, the simulation cost is based on the number of pairwise interactions (n 2 These types of structures in real systems are important and common, but may not be exploited by the most general quantum simulation algorithms.

[0016] Trotterization and subsequent variations, such as the Trotter-Suzuki product formulation, consider very common types of structures and have attracted attention due to their simplicity and low space requirements. For example, they exploit how many terms in the Hamiltonian can commute with each other. This leads to empirical performance of typical quantum systems that is much better than predicted by strict, yet extremely loose, upper bounds. These terms interact only with their neighbors, rather than all of them, thus accounting for both strong and weak interactions based on the distance between them. Consequently, performing these simulations using the product formulation simulation algorithm is significantly less expensive than using worst-case bounds.

[0017] At the first order, the time evolution of the Hamiltonian is approximated by the sequential evolution of its parts, i.e., Therefore, the evolution e for any time t -iHt This is done by applying t / Δ approximation segments, each consisting of N exponents, with Δ chosen to control the overall simulation error. Higher-order variants achieve better scaling, and tight bounds place the number of exponents required using an m-th order integrator at ≤ 2N. 5 m(Nmax j ||h j ||t) 1+1 / m / ∈ 1 / m Due to its simplicity, this approach has already been implemented experimentally and may even find useful applications in the short term, where decoherence still severely limits the number of applied quantum gates.

[0018] The ultimate limit on the performance of Hamiltonian simulations can be obtained by reducing the worst-case computational problem to a physical simulation. Since these bounds indicate that the scaling should be strictly linear in time and logarithmic in error, as The polynomial gap between this formula and any product formula has therefore given rise to a range of post-Trotter simulation algorithms based on different paradigms, such as quantum walks, unitary linear combinations, qubitization, and quantum signal processing, that are able to match this lower bound with respect to all parameters.

[0019] In practice, product formulation methods are simulated for small systems and then extrapolated to larger systems. However, while such Trotter-Suzuki product formulations represent an improvement, they still have limitations, especially when simulating worst-case problems.

[0020] Multi-product formulas were developed in an attempt to address many of the drawbacks of the Trotter-Suzuki formula for solving certain classes of differential equations. The main idea behind them is to use linear combinations of Trotter's formula to construct new integrators for quantum systems. However, most multi-product formulas are ill-conditioned and require exact cancellation of terms to produce reasonable solutions.

[0021] For a quantum circuit implementing a time-evolving operator in time (t), the number of quantum gates required to apply the worst-case algorithm scales linearly in time. The scaling with respect to the error is logarithmic, and the scaling with respect to the number of terms is quadratic. The simplest product formula (e.g., first order) has quadratic scaling in time, cubic scaling with respect to the number of terms, and logarithmic scaling in error (e.g., 1 / error). If the same product formula is used for a long time, the sum of the errors will produce a larger error that is proportional to the simulation time. As the order of the product formula increases, the time and error also increase exponentially with respect to the step size of the simulation. In other words, any m-th order product formula uses a more exponentially large multiplication factor than the post-Trotter simulation algorithm. To approximate the real-time evolution of time t and error ∈, the post-Trotter simulation algorithm achieves the optimal worst-case scaling But the ability to exploit the commutativity of terms is lost.

[0022] Applying very high-order Trotter-Suzuki product formulas to these worst-case problems yields almost linear scaling in time, almost quadratic scaling in the number of terms, and almost logarithmic scaling with respect to the error. This nearly matches the general worst-case algorithm and should yield even better performance for these structured Hamiltonians.

[0023] However, there are also constant prefactors that apply to scaling and cost exponentially. Universal quantum simulation algorithms scale optimally across these parameters and also have very small constant factors for worst-case problems. First-order product formulas scale relatively poorly, but have small constant factors. However, for higher-order Trotter-Suzuki product formulas, the constant factor grows exponentially with the order of the product formula. Therefore, there is a threshold for the order of the product formula above which the absolute time required to run the simulation becomes unfeasibly large. In practice, product formulas of order greater than 4 are rarely used due to this large prefactor.

[0024] Although the product formulations are asymptotically suboptimal for the worst-case problem, their constant factors are significantly smaller than suggested by strict bounds. In fact, numerical studies show that if some Hamiltonian terms are exchanged, the simulation errors are much smaller than expected, to the point where they provide asymptotic advantages. The most striking performance separation is observed when simulating geometrically localized interactions, where even a native application of the product formulation has nearly linear performance in the system size N. In contrast, post-Trotter simulation algorithms lose this desirable feature and, without highly specialized modifications, exhibit quadratic scaling Ω(N 2 ).

[0025] The fact that commutative terms are common in typical quantum systems has stimulated continued interest in product formulas. One approach is based on multi-product formulas, where high-order 2m integrators are constructed from linear combinations of M ≥ m low-order product formulas:

[0026]

[0027] Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, and N is the number of terms. For example, if the product formula is raised to the power of exponent k, then the product formula will be applied k times in sequence. The cost of taking the linear combination of the product formula is the same as the pre-factor a j The error is proportional to the size of Δ. In this example, there are M different product formulas, and therefore M different step sizes (e.g. ). In this example, there are linear combinations of these M different product formulas, where each product formula applies k steps according to the exponential step size. j Second-rate.

[0028] The linear combination of product formulas can be, for example, the symmetric Trotter-Suzuki formula Abbreviated as ||h||=max j ||h|| j . Using the simplest choice k j =j and M=m, only exponent, which seems to be required for the Trotter-Suzuki formula Unfortunately, these multi-product formulas are ill-conditioned - the coefficients (e.g., prefactors) a m =e Ω(mlogm) grows exponentially with the order m. Therefore, even a zero-order constraint In a quantum setting, standard unitary linear combination techniques translate this into an exponentially small probability of success.

[0029] In this paper, we propose a quantum algorithm for Hamiltonian simulation that maintains the average-case properties of product formulas while reducing the worst-case gate complexity from polynomial in t / ∈ to logarithmic. This is achieved by solving the regulation problem faced by multiple product formulas. Sparse solutions are generated by solving the required cancellations using underdetermined systems. This technique can be bootstrapped to any product formula, including classical analogs, while preserving the commutativity of the Hamiltonian terms.

[0030] The more general underdetermined setting of Equation (1) is considered, where M = m, but k j Can be arbitrary, not an arithmetic sequence. The technical result of this solution is that for a size of max j The integer k j , a well-conditioned solution of equation (1) is generated, where becomes exponentially smaller. This can allow the generation and application of higher-order but low-cost product formulas. In practice, this reduces the number of computational operations required to solve quantum computing applications, such as the time evolution of the Hamiltonian operator. Combined with inadvertent amplitude amplification, this allows m-th order integrators to be Cost is deterministically implemented. Therefore, the overhead is reduced to only This translates into the worst-case gate complexity of the Hamiltonian simulation

[0031] Figure 1 An example method 100 for obtaining a solution to an m-order multi-product formula to solve a quantum computing problem including a product formula is shown. The method 100 can be performed by executing stored instructions on a computing device. Such a computing device can be a quantum computing device and / or a classical computing device. The example quantum computing device is described herein with reference to Figure 3 and 4 Describe and exemplify classical computing devices in this article about Figure 5describe.

[0032] At 110, method 100 includes selecting a set of indices k j , where each k j is a real number and is an exponent of a linear combination of product formulas. As an example, the linear combination of product formulas may be equation (1).

[0033] Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function and N is the number of terms. For example, if the product formula is raised to the power of exponent k, then the product formula will be applied k times in sequence. The cost of taking the linear combination of the product formula is the same as the pre-factor a j The error is proportional to the size of Δ. In this example, there are M different product formulas, and therefore M different step sizes (e.g. ). In this example, there are linear combinations of these M different product formulas, where each product formula applies k steps according to the exponential step size. j Second-rate.

[0034] The cost of implementing this linear combination of product formulas is proportional to the 1-norm of the sum of the prefactors and exponents. Method 100 can be used to generate solutions for obtaining a very general class of multiple product formulas where the prefactors a j are all small (e.g., polynomial in size, rather than exponentially large). This product formula can be called a "well-conditioned product formula."

[0035] Theorem — (Hamiltonian simulation via well-conditioned multi-product formulas). There exists a multi-product formula of the form in Eq. 1 The quantum circuit uses the following exponential approximation error at most Real-time evolution operator

[0036]

[0037] in and By minimizing the cost with respect to m,

[0038]

[0039] An explicit construct is also provided, where

[0040] It is noteworthy that the procedure can bootstrap any product formula. Therefore, it preserves the desirable characteristics of product formulas and may similarly exhibit significantly better empirical performance than that suggested by Theorem 1. For example, Table 1 lists tighter bounds showing that

[0041] Multiple product formula

[0042] Any symmetric product formula that approximates the time evolution operator to at least leading order has the following expansion

[0043]

[0044]

[0045] For some error matrix E that is independent of j k and Note that in the Taylor series, Since from e -iHΔ The contribution of has a functional dependence on Δ, but no j dependence.

[0046] By taking a linear combination of these unitary properties described in equation (1), cancellation of all low-order error terms can be designed.

[0047] Return to Figure 1 At 120, method 100 includes the step of: j , based on the underdetermined solution of the m×M linear equation system to determine a set of prefactors a j For example, the following system of linear equations can be generated for the symmetric Trotter-Suzuki formula:

[0048]

[0049] Assume k j is known a priori, the required cancellation can be transformed into this linear system of equations. The left side is the Vandermonde matrix in In the square case M = m, this has the solution

[0050]

[0051] It can be observed that using the arithmetic progression k j =j, coefficient a m =e Ω(mlogm) It is indeed morbid.

[0052] However, Different choices of (not necessarily arithmetic progressions) may produce relatively small coefficients |α j |. In that case, It can be efficiently applied using standard quantum techniques. To illustrate, consider a quantum state preparation circuit and multiplexers The success probability of these applications is Multiple product formula Fortunately, as long as If ∈ is close to unitary, then we can use robust inadvertent amplitude amplification to amplify Each It may be a query of the basic product formula U2-S The cost of the query is then multiplied by the amplification routine Obviously, the performance utilization is smaller value to increase.

[0053] Return to Figure 1 At 130, method 100 includes using a set of indices k j and a set of prefactors a j To solve quantum computing problems involving product formulas.

[0054] k j The values of are real numbers, and are usually integers. Previously, k j The values of are chosen to be an arithmetic progression. Given k j The series, a j The value of can be determined using equation (5). However, this choice of value results in an exponential increase in the coefficient. Using method 100, k j Any number of series can be chosen, and equation (5) can be used to derive the small coefficients for any given progression.

[0055] Fractional query multi-product formula with good conditions

[0056] The above solution is solved by allowing the exponential is an arbitrary real number to relax the conditions of Equation (1). This corresponds to a fractional query of U2, which can be implemented using standard techniques. In this setting, Equation (5) can be mapped to a polynomial interpolation problem.

[0057] For example, consider m polynomials The coefficients of the set are represented as a square matrix These can be chosen to be at some discrete set of interpolation points Orthogonal, that is, and

[0058]

[0059] where δ jk is the Kronecker sign function.

[0060] Now, the Vandermonde matrix can be left-multiplied by the polynomial coefficients A. Therefore, the j-th row satisfies

[0061]

[0062] Using the orthogonality relation in equation (7), this can be obtained by assuming Solve. Therefore, the Chebyshev polynomial basis T j (x) = cos(j cos -1 (x)) can provide multiple sets of product coefficients with desired characteristics.

[0063] Lemma 1: Chebyshev fractional multiproduct

[0064] In equation (5), choose in Then and and

[0065] Proof. Choose the shifted Chebyshev polynomial p j (x) = T j-1 (2x-1) basis set and interpolation, and For example, the Chebyshev polynomial proves

[0066] and p j (0) = -(-1) j Can be used.

[0067] Therefore, the function mean

[0068]

[0069] in and inequalities Already in use.

[0070] The solution is very robust to perturbations. For example, discarding x j The second half of does not affect its qualitative properties, as can be seen in the following results.

[0071] Lemma 2: Semi-Chebyshev Fractional Multiproduct

[0072] In equation (5), choose in Then and Scaling according to Lemma 1.

[0073] Proof. Using 2m interpolation points Bootstrapping the solution in Lemma 1, which has the multi-product coefficients given by Eq. (9) Using the exact solution of equation (6), we can observe that

[0074]

[0075] For all 1≤j<q≤2m, this is determined by Produced. Similar to Lemma 1, the number and can be evaluated.

[0076] Well-conditioned integer query multi-product formula

[0077] Lemmas 1 and 2 choose k j is a real number but not necessarily an integer. This generally shows that using real numbers, equation (5) can be solved to obtain very small coefficients. Since the sum of the exponent and the coefficient is small, their product is also small, so a higher order product formula can be used at a lower cost. However, for k j The solution using real numbers may be impractical to implement. However, this provides a proof of principle that if you can use k j The cost can be reduced if .

[0078] It can now be shown that by rescaling with a suitable constant K and converting each exponential Round to different integers l j -otherwise To become singular, fractional query solutions can be converted to integer query solutions. This bootstrapping does not significantly increase the cost and still produces polynomial scaling of the coefficients with respect to the order.

[0079] Rounding also messes up multi-product coefficients But similar to the robustness of Lemma 2, at most the multiplication constant is changed, as shown below.

[0080] Lemma 3: Half-Chebyshev integer multiplication

[0081] In equation (5), choose in x j From Lemma 2, and K∈Θ(m). Then, and

[0082] Proof. By x j The definition of k j ∈Θ(Km / j) and K∈Θ(m), so that l j and k j Round to a unique integer. Therefore, the perturbation |lj -k j |=δ j ∈Θ(K / m) and γ q =(k q / k j ) 2 The score change is

[0083]

[0084] where the sign of Δ[q,j] matches the sign of (qj).

[0085] and original The change between can now be evaluated. Since Δ[q, j] is small, the change to the leading order The shift can be achieved by using the derivative To evaluate.

[0086] therefore

[0087]

[0088] In the last line, the asymptotic expression is expressed in terms of x j , x q replace.

[0089] Because the index k j has been chosen a priori, so if Minimized to all Subset Then the smaller 1-norm It may be possible. Surprisingly, the exact solution to this discrete optimization problem can be obtained efficiently via a linear program

[0090]

[0091] Then it is minimized with respect to m∈[M]. Importantly, minimizing the 1-norm also ensures that we know There are exactly m non-zero elements, similar to sparse signal recovery in compressed sensing. In this context, "minimization" refers to a procedure for reducing the relative values and / or sum of the components, not necessarily deriving a global minimum.

[0092] Equation (5) is traditionally chosen to be a square matrix, which means that the free parameters (k j ) is exactly equal to the order of the integrator. In this paper, Equation (5) is solved as an underdetermined system, which means that there are extra free parameters. In other words, k jThe value of can be more than the integrator order M. By solving the underdetermined system, using the clause of the optimization problem in equation (13), a sparse solution can be derived. As an example, the sparse solution includes as many a as possible j Value, will = 0. Additionally or alternatively, a j A subset of the values is set to zero and / or small values. Thus, even if there is an arithmetic progression in the Vandermonde matrix in Eq. (5), in some examples all coefficients that can be zero are set to zero, assuming that the 1-norm of these coefficients is as small as possible. By making the solution sparse, the cost of the algorithm also depends on k j 1-norm, but if a j The coefficient of is 0, then the corresponding k j is not applied, and the 1-norm of k is just the sum of the positive a j k associated with the value j The sum of the values.

[0093] Worst-case error bound

[0094] Tight but loose bounds can now be provided on the error of the multi-product integrator, completing the proof of Theorem 1.

[0095] Proof of Theorem 1. By Represents the remainder of any function.

[0096] Observe the remainder of the product formula

[0097]

[0098] has an error bound independent of the index j, where And the inequality and Has been used.

[0099] Therefore, a single multi-product segment has an error

[0100]

[0101] By applying r≤1 / ∈ t / r segment and use inequalities The error of the simulation for a longer time t is

[0102] For example, select Now use the solution log(tλ / ∈)=(z)e z The Lambert-W function is defined as z = W (log (tλ / ∈)), and 2m + 1 = e z Implementation The total number of exponents required is then

[0103]

[0104] in Scaling according to Lemma 3.

[0105] Figure 2 Graph 200 is shown comparing the relative costs of the classic Trotter-Suzuki product formula and various implementations of the method described herein. Graph 200 plots the number of queries of the second-order Trotter-Suzuki formula U2 for each step of the 2m-order integrator. Plot 210 represents the Trotter-Suzuki formula. Plot 220 represents the rounded Chebyshev method of Lemma 3. Plot 230 represents the linear program. Plot 240 represents the fractional Chebyshev method of Lemma 1. As expected, the fractional query solution of plot 240 has a very small cost, but can be more complex to implement. The rounded Chebyshev solution of plot 220 is generated by bootstrapping the fractional query solution. While not as cost-effective as the linear program of plot 230, this rounded Chebyshev solution significantly reduces the cost compared to the Trotter-Suzuki. For very small systems—say, 15 qubits—minimizing cost allows the use of integrators of approximately order 10. As the size of the computing system increases, the order of the integrator can be increased accordingly.

[0106] These theoretical solutions are scaled similarly to the minimized solutions (i.e., solutions that are the result of a minimization process, which may or may not be actual minima) presented in Table 1. Table 1 presents the minimized The multi-product solution of equation (1) is Only include the values corresponding to non-zero a j Coefficient k j Exponential. Both theoretical and practical solutions allow for the generation and application of higher-order, yet inexpensive, product formulas for solving quantum computing applications, such as the time evolution of the Hamiltonian operator. The technical benefits of implementing these solutions include reducing the number of computational operations required to generate a solution, thereby reducing the time and computing power required to perform quantum simulations.

[0107] Table 1

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114] Although primarily described with respect to symmetric product formulas (such as the second-order symmetric Trotter-Suzuki product formula), similar methods can be applied to other types of product formulas. For example, the extended variations shown in Equation (4) may produce variations of the Vandermonde matrix shown in Equation (5). The solutions to these variations can be used to generate efficient multiple product formulas.

[0115] For example, when the underlying product formula is symmetric and of order α rather than 2, a slightly different system of equations can be solved:

[0116]

[0117] When the basic product formula is nonsymmetric and of order 1, the following system of equations can be solved:

[0118]

[0119] When the basic product formula is nonsymmetric and of order α, the following system of equations can be solved:

[0120]

[0121] As described for Equation (5), underdetermined solutions to Equations (17 to 19) can be generated such that k j It does not have to be an arithmetic series (e.g. using equation (13)). By choosing the exponent k more flexibly j , a well-conditioned multi-product formula with the desired properties can be generated, where the coefficients a j The sum of the absolute values is relatively small.

[0122] In some embodiments, the methods and processes described herein may be associated with a quantum computing system of one or more quantum computing devices. Figure 4 Various aspects of an example quantum computer 310 configured to perform quantum logic operations (see below) are shown. Whereas conventional computer memory stores digital data in arrays of bits and performs bit-by-bit logic operations, a quantum computer stores data in arrays of qubits and performs quantum mechanical operations on the qubits to implement the desired logic. Thus, Figure 3 Quantum computer 310 includes at least one register 312 comprising a qubit array 314. The illustrated register is 8 qubits in length; registers comprising longer and shorter qubit arrays are also contemplated, as are quantum computers comprising two or more registers of any length.

[0123] The qubits of register 312 can take various forms, depending on the desired architecture of quantum computer 310. Each qubit 314 can include, as non-limiting examples, a superconducting Josephson junction, a trapped ion, a trapped atom coupled to a high-finesse cavity, an atom or molecule confined within a fullerene, an ion or neutral dopant atom confined within a host lattice, a quantum dot exhibiting discrete spatial or spin electronic states, an electron-hole in a semiconductor junction entrained via an electrostatic trap, a coupled quantum wire pair, an atomic nucleus addressable by magnetic resonance, a free electron in helium, a molecular magnet, or a metallic carbon nanosphere. More generally, each qubit 314 can include any particle or particle system capable of existing in two or more discrete quantum states that can be experimentally measured and manipulated. For example, qubits can also be implemented in multiple processing states corresponding to different modes of light propagation through linear optical elements (e.g., mirrors, beam splitters, and phase shifters), as well as in states accumulated within a Bose-Einstein condensate.

[0124] Figure 4 is an illustration of a Bloch sphere 316, which provides a graphical description of some quantum mechanical aspects of an individual qubit 314. In this description, the north and south poles of the Bloch sphere correspond to the standard basis vectors |0> and |1>, respectively, for electrons or other fermions—for example, spin-up and spin-down states. The set of points on the surface of the Bloch sphere includes all possible pure states |ψ> of the qubit, while the interior points correspond to all possible mixed states. Mixed states for a given qubit can arise from decoherence, which can occur due to undesired coupling with external degrees of freedom.

[0125] Now return to Figure 3 , quantum computer 310 includes a controller 318. The controller may include conventional electronic components, including at least one processor 320 and an associated memory device 322. The term 'conventional' is applied herein to any component that can be modeled as a collection of particles, without regard to the quantum state of any individual particle. For example, conventional electronic components include integrated, microlithographic transistors, resistors, and capacitors. The memory device 322 may be configured to store program instructions 324 that cause the processor 320 to perform any of the processes described herein. Additional aspects of the controller 318 are described below.

[0126] Controller 318 of quantum computer 310 is configured to receive a plurality of inputs 326 and provide a plurality of outputs 328. The inputs and outputs may comprise digital and / or analog lines, respectively. At least some of the inputs and outputs may be data lines, through which data is supplied to and retrieved from the quantum computer. Other inputs may include control lines, via which the operation of the quantum computer may be adjusted or otherwise controlled.

[0127] Controller 318 is operably coupled to register 312 via interface 330. The interface is configured to bidirectionally exchange data with the controller. The interface is also configured to bidirectionally exchange signals corresponding to the data with the register. Depending on the architecture of quantum computer 310, such signals may include electrical signals, magnetic signals, and / or optical signals. Via signals conveyed through the interface, the controller can interrogate and otherwise influence the quantum state stored in the register, as defined by the collective quantum state of qubit array 314. To this end, the interface includes at least one modulator 332 and at least one demodulator 334, each operably coupled to one or more qubits of register 312. Each modulator is configured to output a signal to the register based on modulated data received from the controller. Each demodulator is configured to sense a signal from the register and output data to the controller based on the signal. In some scenarios, the data received from the demodulator may be an estimate of a measured observable of the quantum state stored in the register.

[0128] More specifically, a suitably configured signal from the modulator 332 can physically interact with one or more qubits 314 of the register 312 to trigger a measurement of the quantum state held in the one or more qubits. The demodulator 334 can then sense the resulting signal released by the one or more qubits in response to the measurement and can provide data corresponding to the resulting signal to the controller. In other words, the demodulator can be configured to reveal an estimate of an observable reflecting the quantum state of the one or more qubits of the register based on the received signal and provide the estimate to the controller 318. In one non-limiting example, the modulator can provide an appropriate voltage pulse or sequence of pulses to the electrodes of the one or more qubits based on the data from the controller to initiate the measurement. In the short term, the demodulator can sense the emission of photons from the one or more qubits and can maintain a corresponding digital voltage level on the interface line into the controller. In general, any measurement of a quantum mechanical state is performed by an operator corresponding to the observable to be measured. Definition: The measurement result R is guaranteed to be In quantum computer 310, R is statistically related to the register state before measurement, but is not uniquely determined by the register state.

[0129] Based on appropriate input from controller 318, interface 330 can also be configured to implement one or more quantum logic gates to operate on the quantum state stored in register 312. While the function of each type of logic gate in a conventional computer system is described by a corresponding truth table, the function of each type of quantum gate is described by a corresponding operator matrix. The operator matrix operates on (i.e., multiplies) the complex vector representing the register state and implements a specified rotation of the vector in Hilbert space.

[0130] exist Figure 3 Continuing in, appropriately configured signals from modulator 332 of interface 330 can physically interact with one or more qubits 314 of register 312 in order to maintain any desired quantum gate operation. As mentioned above, the desired quantum gate operation is a specifically defined rotation of the complex vector representing the register state. To achieve the desired rotation One or more modulators of interface 330 may be switched on for a predetermined duration T i Apply a predetermined signal level S i .

[0131] In some examples, multiple signal levels can be applied for multiple sequences or otherwise related durations. In more specific examples, multiple signal levels and durations are arranged to form a composite signal waveform that can be applied to one or more qubits of a register. Typically, each signal level S i and each duration T i is a control parameter adjustable by appropriate programming of the controller 318. In other quantum computing architectures, a different set of adjustable control parameters may control the quantum operations applied to the register state.

[0132] In some embodiments, the methods and processes described herein can be associated with a classical computing system of one or more classical computing devices. Specifically, such methods and processes can be implemented as a computer application or service, an application programming interface (API), a library, and / or other computer program products.

[0133] Figure 5 A non-limiting embodiment of a computing system 500 capable of enacting one or more of the above-described methods and processes is schematically illustrated. Computing system 500 is shown in simplified form. Computing system 500 can take the form of one or more personal computers, server computers, tablet computers, home entertainment computers, network computing devices, gaming devices, mobile computing devices, mobile communication devices (e.g., smartphones), and / or other computing devices.

[0134] The computing system 500 includes a logic machine 510 and a storage machine 520. The computing system 500 may optionally include a display subsystem 530, an input subsystem 540, a communication subsystem 550, and / or Figure 5 Other components not shown.

[0135] The logic engine 510 includes one or more physical devices configured to execute instructions. For example, the logic engine can be configured to execute instructions that are part of one or more applications, services, programs, routines, libraries, objects, components, data structures, or other logical constructs. Such instructions can be implemented to perform tasks, implement data types, transform the state of one or more components, achieve technical effects, or otherwise achieve a desired result.

[0136] The logic machine may include one or more processors configured to execute software instructions. Additionally or alternatively, the logic machine may include one or more hardware or firmware logic machines configured to execute hardware or firmware instructions. The processors of the logic machine may be single-core or multi-core, and the instructions executed thereon may be configured for sequential, parallel, and / or distributed processing. The individual components of the logic machine may optionally be distributed across two or more separate devices that may be remotely located and / or configured for coordinated processing. Various aspects of the logic machine may be virtualized and executed by a remotely accessible, networked computing device configured in a cloud computing configuration.

[0137] The storage machine 520 includes one or more physical devices configured to store instructions executable by the logic machine to implement the methods and processes described herein. When such methods and processes are implemented, the state of the storage machine 520 may be transformed, for example, to store different data.

[0138] The storage device 520 may include removable and / or built-in devices. The storage device 520 may include optical storage (e.g., CD, DVD, HD-DVD, Blu-ray disc, etc.), semiconductor storage (e.g., RAM, EPROM, EEPROM, etc.), and / or magnetic storage (e.g., hard disk drive, floppy disk drive, tape drive, MRAM, etc.). The storage device 520 may include volatile, non-volatile, dynamic, static, read / write, read-only, random access, sequential access, location addressable, file addressable, and / or content addressable devices.

[0139] It is understood that storage 520 comprises one or more physical devices. Alternatively, however, various aspects of the instructions described herein may be propagated by a communication medium (eg, electromagnetic signals, optical signals, etc.) that is not retained for a finite duration by a physical device.

[0140] Various aspects of the logic engine 510 and the memory engine 520 may be integrated together into one or more hardware logic components. For example, such hardware logic components may include field programmable gate arrays (FPGAs), program and application specific integrated circuits (PASIC / ASIC), program and application specific standard products (PSSP / ASSP), systems on chips (SOCs), and complex programmable logic devices (CPLDs).

[0141] When included, the display subsystem 530 can be used to present a visual representation of the data stored by the storage machine 520. This visual representation can take the form of a graphical user interface (GUI). Since the methods and processes described herein change the data stored by the storage machine and thus transform the state of the storage machine, the state of the display subsystem 530 can likewise be transformed to visually represent the changes in the underlying data. The display subsystem 530 can include one or more display devices utilizing virtually any type of technology. Such a display device can be combined with the logic machine 510 and / or the storage machine 520 in a shared enclosure, or such a display device can be a peripheral display device.

[0142] When included, the input subsystem 540 may include or interface with one or more user input devices, such as a keyboard, mouse, touch screen, or game controller. In some embodiments, the input subsystem may include or interface with selected natural user input (NUI) component portions. Such component portions may be integrated or peripheral, and the transduction and / or processing of input actions may be handled on-board or off-board. Example NUI component portions may include microphones for voice and / or sound recognition; infrared, color, stereo, and / or depth cameras for machine vision and / or gesture recognition; head trackers, eye trackers, accelerometers, and / or gyroscopes for motion detection and / or intent recognition; and electric field sensing component portions for assessing brain activity.

[0143] When included, the communication subsystem 550 can be configured to communicatively couple the computing system 500 with one or more other computing devices. The communication subsystem 550 can include wired and / or wireless communication devices compatible with one or more different communication protocols. As non-limiting examples, the communication subsystem can be configured to communicate via a wireless network or a wired or wireless local area network or wide area network. In some embodiments, the communication subsystem can allow the computing system 500 to send messages to and / or receive messages from other devices via a network such as the Internet.

[0144] In one example, a method for obtaining a solution to an m-order multi-product formula to solve a quantum computing problem including a product formula, via execution of stored instructions on a computing device, includes: selecting a set of exponents k j , where each k j is a real number and is the exponent of a linear combination of product formulas; based on a set of exponents k j , based on the underdetermined solution of the m×M linear equation system to determine a set of prefactors a j , where the exponential is expressed as a Vandermonde matrix where α is the order of the product formula, and where M is the number of lower-order product formulas included in the linear combination of product formulas; and using a set of exponents k j and a set of prefactors a j To solve quantum computing problems involving product formulas. In this example or any other example, the m×M linear equations are additionally or alternatively expressed as: wherein the quantum computation problem is additionally or alternatively a Hamiltonian simulation, and wherein the linear combination of product formulas is additionally or alternatively as follows: Where U represents the product formula, Δ is the step size, a j is a set of prefactors, H is the sum of Hamiltonian operators h, is a cost function, and N is the number of terms. In any of the foregoing examples or any other examples, the linear combination of the product formulas is additionally or alternatively an expression of the symmetric Trotter-Suzuki formula, as follows: In any of the foregoing examples, or any other examples, where α=2, the Vandermonde matrix is additionally or alternatively as follows: In any of the preceding examples or any other examples, where M=m, the underdetermined solution of the m×M linear system of equations additionally or alternatively has the solution: In any of the foregoing examples or any other examples, the underdetermined solution to the m×M system of linear equations is additionally or alternatively determined based on the following linear program: And k j =j. In any of the foregoing examples or any other examples, the underdetermined solution of the m×M linear system of equations is additionally or alternatively based on a fractional query on U2, and wherein the index k j is any real number. In any of the foregoing examples or any other examples, the underdetermined solution of the m×M linear system of equations is additionally or alternatively determined based on fractional multiple products, where is chosen so that

[0145] In another example, a computing device includes: a processor; and a storage device storing instructions, in response to a request to obtain a solution to an m-order multi-product formula to solve a quantum computing problem including a product formula, the instructions causing the processor to: select a set of exponents k j , where each k j is a real number and is the exponential of the linear combination of the following product formulas: Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, N is the number of terms; based on a set of exponential kj , determine a set of prefactors a based on the underdetermined solution of the following m×M linear equations j : And using a set of indices k j and a set of prefactors a j To solve a quantum computing problem involving a product formula. In this or any other example, the linear combination of the product formula is additionally or alternatively a symmetric Trotter-Suzuki formula: In any of the preceding examples or any other examples, where M=m, the underdetermined solution of the m×M linear system of equations additionally or alternatively has the solution: In any of the foregoing examples or any other examples, the underdetermined solution to the m×M system of linear equations is additionally or alternatively determined based on the following linear program: And k j =j. In any of the foregoing examples or any other example, the underdetermined solution of the m×M linear system of equations is additionally or alternatively based on a fractional query on U2, and the index k j Additionally or alternatively is any real number. In any of the foregoing examples or any other examples, the underdetermined solution of the m×M linear system of equations is additionally or alternatively determined based on fractional multiple products, where Additionally or alternatively is selected so that In any of the foregoing examples or any other examples, the computing device is additionally or alternatively a quantum computing device and further includes: a register comprising a plurality of qubits; a modulator configured to perform a quantum logic operation on the plurality of qubits according to control parameter values stored at a storage device; and a demodulator configured to reveal data reflecting a quantum state of the plurality of qubits; and wherein the controller is operably coupled to the modulator and the demodulator. In any of the foregoing examples or any other examples, the computing device is additionally or alternatively a classical computing device.

[0146] In yet another aspect, a quantum computing device implemented method for simulating the time evolution of a Hamiltonian operator via an m-order multi-product formula includes: selecting a set of exponents k j , where each k j is a real number and is the exponential of the linear combination of the following product formulas: Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, N is the number of terms; based on a set of exponential k j , determine a set of prefactors a based on the underdetermined solution of the following m×M linear equationsj : The underdetermined solution is based on the following linear program: And k j = j; and using a set of indices k j and a set of prefactors a j To simulate the time evolution of the Hamiltonian operator. In this example or any other example, where M=m, the underdetermined solution of the m×M linear equations additionally or alternatively has the solution: In any of the foregoing examples or any other examples, the underdetermined solution of the m×M linear system of equations is additionally or alternatively based on a fractional query on U2, and wherein the index k j Additionally or alternatively is any real number. In any of the foregoing examples or any other examples, the linear combination of product formulas is additionally or alternatively a symmetric Trotter-Suzuki formula:

[0147] It is to be understood that the configurations and / or methods described herein are exemplary in nature, and these specific embodiments or examples should not be considered restrictive, as many variations are possible. The specific routines or methods described herein can represent one or more of any number of processing strategies. Therefore, the various actions illustrated and / or described can be performed in the illustrated and / or described sequence, in other sequences, in parallel, or omitted. Similarly, the order of the above-mentioned processes can be changed.

[0148] The subject matter of the present disclosure includes all novel and non-obvious combinations and sub-combinations of the various processes, systems and configurations, and other features, functions, acts, and / or properties disclosed herein, as well as any and all equivalents thereof.

Claims

1. A method for simulating the time evolution of a Hamiltonian operator of a quantum system via an m-order multi-product formula, the method being formulated via execution of stored instructions on a computing device, the method comprising: Choose a set of indices k j , where each k j is a real number and is an exponent of a linear combination of product formulas; Based on the set of indices k j , based on the underdetermined solution of the m×M linear equation system to determine a set of prefactors a j , where the exponential is expressed as a Vandermonde matrix wherein α is the order of the product formula, and wherein M is the number of lower-order product formulas in the linear combination of the product formulas, wherein m≠M; as well as Using the set of indices k j and the set of pre-factors a j To solve the quantum computing problem involving the product formula, The m×M linear equations are expressed as: The quantum computing problem is a Hamiltonian simulation, and the linear combination of the product formulas is as follows: Where U represents the product formula, Δ is the step size, a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, and N is the number of terms.

2. The method according to claim 1, wherein the linear combination of the product formula is an expression of the symmetric Trotter-Suzuki formula, as follows:

3. The method of claim 1, wherein α=2, and wherein the Vandermonde matrix is as follows:

4. The method of claim 1 , wherein the underdetermined solution to the m×M system of linear equations is determined based on the following linear programming: And k j =j.

5. The method of claim 4, wherein the underdetermined solution to the m×M linear system of equations is based on a fractional query on U2, and wherein the index k j is any real number.

6. The method of claim 4, wherein the underdetermined solution to the m×M linear system of equations is determined based on fractional multiplication, wherein is chosen so that 7. A computing device comprising: processor; as well as A storage device storing instructions that, when executed by the processor, cause the processor to simulate a time evolution of a Hamiltonian operator of a quantum system via an m-order multi-product formula, the instructions causing the processor to: Choose a set of indices k j , where each k j is a real number and is the exponential of a linear combination of the following product formulas: Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of the product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, N is the number of terms; Based on the set of indices k j , the set of prefactors a is determined based on the underdetermined solution of the following m×M linear equations j , where m≠M: as well as Using the set of indices k j and the set of pre-factors a j To solve quantum computing problems involving the product formula.

8. The computing device of claim 7, wherein the linear combination of product formulas is a symmetric Trotter-Suzuki formula:

9. The computing device of claim 8, wherein the underdetermined solution to the m×M system of linear equations is determined based on the following linear programming: And k j =j.

10. The computing device of claim 8, wherein the underdetermined solution to the m×M system of linear equations is based on a fractional query on U2, and wherein the index k j is any real number.

11. The computing device of claim 8, wherein the underdetermined solution to the m×M system of linear equations is determined based on fractional multiple products, wherein is chosen so that 12. The computing device of claim 7, wherein the computing device is a quantum computing device, the computing device further comprising: a register, comprising a plurality of qubits; a modulator configured to perform a quantum logic operation on the plurality of qubits according to control parameter values stored at the storage device; as well as a demodulator configured to reveal data reflecting the quantum state of the plurality of qubits; and wherein a controller is operatively coupled to the modulator and the demodulator.

13. The computing device of claim 7, wherein the computing device is a classical computing device.

14. A method implemented by a quantum computing device for simulating the time evolution of a Hamiltonian operator via an m-order multi-product formula, the method comprising: Choose a set of indices k j , where each k j is a real number and is the exponential of a linear combination of the following product formulas: Where U represents the product formula, M is the number of low-order product formulas included in the linear combination of the product formulas, Δ is the step size, and a j is a set of prefactors, H is the sum of Hamiltonian operators h, is the cost function, N is the number of terms; Based on the set of indices k j , the set of prefactors a is determined based on the underdetermined solution of the following m×M linear equations j , where m≠M: The underdetermined solution is based on the following linear program: And k j =j; and Using the set of indices k j and the set of pre-factors a j to simulate the time evolution of the Hamiltonian operator.

15. The method implemented by a quantum computing device according to claim 14, wherein the underdetermined solution of the m×M linear system of equations is based on a fractional query on U2, and wherein the exponent k j is any real number.

16. The method implemented by a quantum computing device according to claim 14, wherein the linear combination of the product formulas is an expression of the symmetric Trotter-Suzuki formula, as follows:

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