Publication number | WO2016011440 A1 |

Publication type | Application |

Application number | PCT/US2015/041090 |

Publication date | 21 Jan 2016 |

Filing date | 20 Jul 2015 |

Priority date | 18 Jul 2014 |

Publication number | PCT/2015/41090, PCT/US/15/041090, PCT/US/15/41090, PCT/US/2015/041090, PCT/US/2015/41090, PCT/US15/041090, PCT/US15/41090, PCT/US15041090, PCT/US1541090, PCT/US2015/041090, PCT/US2015/41090, PCT/US2015041090, PCT/US201541090, WO 2016/011440 A1, WO 2016011440 A1, WO 2016011440A1, WO-A1-2016011440, WO2016/011440A1, WO2016011440 A1, WO2016011440A1 |

Inventors | Alan Aspuru-Guzik, Salvatore MANDRA, Gian Giacomo GUERRESCHI |

Applicant | President And Fellows Of Harvard College |

Export Citation | BiBTeX, EndNote, RefMan |

Patent Citations (3), Classifications (3), Legal Events (3) | |

External Links: Patentscope, Espacenet | |

WO 2016011440 A1

Abstract

An approach to controlling a quantum computer provides a technical advantage by reducing the computation time. Such shorter computation time is obtained by controlling the schedule that determines the instantaneous Hamiltonian for the quantum computer. The approach uses characterization of a problem and driver Hamiltonian that allows computationally efficient calculation of the schedule, which is based on reducing the effective dimensionality of the quantum system.

Claims (OCR text may contain errors)

1. A method for controlling a quantum computer to solve a problem, the method comprising:

accepting a first data representation of a problem Hamiltonian corresponding to a final quantum state of the quantum computer, where a measurement of the final quantum state at an end time provides at least one solution to the problem;

accepting a second data representation of a driver Hamiltonian corresponding to an initial quantum state of the quantum computer at a start time;

applying a computer-implemented procedure to determine a schedule function that determines an interpolation of a total Hamiltonian from the driver Hamiltonian to the problem Hamiltonian at multiple times between the start time and the end time, the procedure including:

forming a third data representation of the total Hamiltonian from a combination of the first data representation and the second data representation,

determining multiple states from the third data representation, forming an overlap matrix from the multiple states, the overlap matrix effectively defining a Hilbert space, and

determining a spectrum of the total Hamiltonian restricted to a subspace of the Hilbert space, where the dimensionality of the subspace is smaller than the dimensionality of the Hilbert space; and controlling a time evolution of a configuration of the quantum computer between the start time and the end time according to the determined schedule function.

2. The method of claim 1 , wherein the procedure further includes forming a block diagonal matrix from the overlap matrix.

3. The method of claim 2, wherein forming a block diagonal matrix from the overlap matrix includes transforming a basis of the overlap matrix.

4. The method of claim 2, wherein the subspace corresponds to a selected block of the block diagonal matrix.

5. The method of claim 4, wherein the block diagonal matrix contains only a single block that is larger than 1 x 1 in size, which is the selected block.

6. The method of claim 4, wherein determining the spectrum of the subspace includes determining eigenvalues of the selected block.

7. The method of claim 6, wherein the procedure further includes determining, based on the determined spectrum, a spectral gap at each of the multiple times between the start time and the end time, the spectral gap at a particular time corresponding to a difference between a ground state energy and a first exited state energy of an effective Hamiltonian corresponding to the selected block.

8. The method of claim 7, wherein controlling the time evolution of the configuration of the quantum computer between the start time and the end time according to the determined schedule function includes varying the total Hamiltonian more slowly when the spectral gap is smaller and more quickly when the spectral gap is larger.

9. The method of claim 1 , wherein the third data representation includes: (1) a first term representing a first Hamiltonian that is highly degenerate with M distinct energy levels, and (2) a second term representing a second Hamiltonian that is a sum of k projectors on k respective pure quantum states.

10. The method of claim 9, wherein at least one of the first Hamiltonian or the second Hamiltonian includes contributions from both the driver Hamiltonian and the problem Hamiltonian.

1 1. The method of claim 10, wherein both of the first Hamiltonian and the second Hamiltonian include contributions from both the driver Hamiltonian and the problem Hamiltonian.

12. The method of claim 9, wherein determining multiple states from the third data representation includes determining k M states corresponding to the k pure quantum states of the second Hamiltonian projected onto each of multiple eigensubspaces of the first Hamiltonian.

13. The method of claim 9, wherein the Hilbert space has a dimensionality of 2^{N} and the subspace has a dimensionality of / M , where k and / and M are integers and /≤ k .

14. The method of claim 1 , wherein a quantum state representing the quantum computer and subject to the total Hamiltonian undergoes adiabatic dynamics that are confined to the subspace.

15. A quantum computing system comprising a scheduling computer configured to control a quantum computer by performing all the steps of any of claims 1 to 14.

16. Software comprising instructions stored on a non-transitory computer-readable medium for causing a scheduling computer to control a quantum computer by performing all the steps of any of claims 1 to 14.

Description (OCR text may contain errors)

QUANTUM P RO C E S S O R S CHEDULE C ONTRO L

Cross-reference to Related Applications

[001] This application claims the benefit of U.S. Provisional Application No.

62/026,090 filed on July 18, 2014, which is incorporated herein by reference.

Statement as to Federally Sponsored Research

[002] This invention was made with government support under contract number FA9550-12-1-0046 awarded by the United States Air Force Office of Scientific Research and contract number DE-SC0001088 awarded by the Department of Energy. The government has certain rights in the invention.

Background

[003] This description relates to the solution of problems (e.g., binary optimization problems) by using quantum computers, and more particularly to controlling a quantum computer according to an evolution schedule.

[004] Approaches described in this document primarily focus on analogic adiabatic quantum computation. An example of such an approach, and a physical realization of a quantum computer to host such an approach, is described in US Pat. Pub. 2010/0306142, titled "Methods for Adiabatic Quantum Computation." This publication, is incorporated herein by reference.

[005] The class of binary optimization problems referred to as Polynomial Unconstrained Binary Optimization (PUBO) consists in finding an optimal assignment y of N bits, where such that y minimizes a certain N-bit cost function

f (x) , with namely y = argmin_{i} f(x) . The cost function f (x) has a

unique representation as:

[006] A cost function that receives an N-bit configuration as input can be written according to the above expression. The coefficients c_{s} completely specify the cost function / .
[007] Many problems that are not initially formulated as optimization problems can be rephrased in a form compatible with the minimization of a cost function. In this case as well, the solution of the optimization problem is the N-bit configuration that minimizes

[008] The coefficients c_{s} also specify a "problem" Hamiltonian H_{p} (f) , used to configure the quantum computer (namely the physical machine). In many common implementations and configurations of a quantum computer, each problem variable x_{i} corresponds to a physical element (usually a quantum two-level system) of the physical machine, typically referred to as a qubit. In some implementations, a single variable x_{i} is associated with several physical qubits.

[009] Without loss of generality, H_{p}(f) can always be expressed as

[010] where | x) represents the quantum state associated with the N-bit configuration x .There is a one-to-one correspondence between the cost function / and the problem Hamiltonian H_{p}(f) . Accordingly, the problem is solved when the state corresponding to the ground state (namely the lowest energy state) of H_{p}{f) is produced and measured.

[011] Adiabatic quantum computation (AQC) is based on the adiabatic quantum theorem: if a system is initially in the ground state of a certain Hamiltonian that is later changed continuously in time, the system will dynamically evolve to be, at any instant, in the ground state of the instantaneous Hamiltonian. This is correct only if the Hamiltonian is changed in time slowly enough to be considered adiabatic.

[012] Several criteria have been proposed to ensure that the evolution is adiabatic, i.e. to verify that the Hamiltonian is changed slowly enough. The relevant conditions for fulfilling these criteria are described below.

[013] AQC consists in slowly changing the following time-dependent Hamiltonian

with H_{D} typically referred to as the "driver" Hamiltonian, for which the ground state is known and its preparation is practical on the physical machine. H_{D} also has the property of not commuting with H_{p} .
[014] It should be understood that the techniques described in this document are not limited to a particular choice of H_{D} .

[015] The adiabatic schedule function s(t) is an arbitrary function such that s(0) = 0 and s(T) = 1 , where T is the total evolution time. The schedule function s(t) controls the quantum computer, and, together with H_{D} and H_{p} , determines the probability of success of the computation.

[016] The evolution time T (which is given by s^{-1}(1) ) determines how long the quantum computation takes to complete the adiabatic optimization.

[017] On one hand, the evolution time T should be chosen as small as possible to reduce the adiabatic computational time. On the other hand, T should be selected to be sufficiently long so that the quantum computer evolution is adiabatic and it is therefore ensured that the system remains in its instantaneous ground state at any time.

[018] A common choice consists in using a linear schedule where the

evolution time T is chosen to be proportional to The value g_{min} is problem

dependent and it is defined as the minimum energy gap between the ground state and the first excited state of H_{AQC} during the whole adiabatic computation.

[019] Another choice, for example adopted by the D-wave machines (as described in Johnson et at. Nature, 473(7346): 194-198, May 201 1 , incorporated herein by reference), is to follow a non-linear schedule that is problem independent, but specific to the hardware.

[020] Both procedures described in the two previous paragraphs have proven to be not optimal in general.

Summary

[021] In one aspect, in general, a method for controlling a quantum computer to solve a problem includes: accepting a first data representation of a problem Hamiltonian corresponding to a final quantum state of the quantum computer, where a measurement of the final quantum state at an end time provides at least one solution to the problem; accepting a second data representation of a driver Hamiltonian corresponding to an initial quantum state of the quantum computer at a start time; applying a computer- implemented procedure to determine a schedule function that determines an interpolation of a total Hamiltonian from the driver Hamiltonian to the problem Hamiltonian at multiple times between the start time and the end time, the procedure including: forming a third data representation of the total Hamiltonian from a combination of the first data representation and the second data representation, determining multiple states from the third data representation, forming an overlap matrix from the multiple states, the overlap matrix effectively defining a Hilbert space, and determining a spectrum of the total Hamiltonian restricted to a subspace of the Hilbert space, where the dimensionality of the subspace is smaller than the dimensionality of the Hilbert space; and controlling a time evolution of a configuration of the quantum computer between the start time and the end time according to the determined schedule function.

[022] Aspects can include one or more of the following features.

[023] The procedure further includes forming a block diagonal matrix from the overlap matrix.

[024] Forming a block diagonal matrix from the overlap matrix includes transforming a basis of the overlap matrix.

[025] The subspace corresponds to a selected block of the block diagonal matrix.

[026] The block diagonal matrix contains only a single block that is larger than 1 x 1 in size, which is the selected block.

[027] Determining the spectrum of the subspace includes determining eigenvalues of the selected block.

[028] The procedure further includes determining, based on the determined spectrum, a spectral gap at each of the multiple times between the start time and the end time, the spectral gap at a particular time corresponding to a difference between a ground state energy and a first exited state energy of an effective Hamiltonian corresponding to the selected block.

[029] Controlling the time evolution of the configuration of the quantum computer between the start time and the end time according to the determined schedule function includes varying the total Hamiltonian more slowly when the spectral gap is smaller and more quickly when the spectral gap is larger. [030] The third data representation includes: (1) a first term representing a first Hamiltonian that is highly degenerate with M distinct energy levels, and (2) a second term representing a second Hamiltonian that is a sum of k projectors on k respective pure quantum states.

[031] At least one of the first Hamiltonian or the second Hamiltonian includes contributions from both the driver Hamiltonian and the problem Hamiltonian.

[032] Both of the first Hamiltonian and the second Hamiltonian include contributions from both the driver Hamiltonian and the problem Hamiltonian.

[033] Determining multiple states from the third data representation includes determining k M states corresponding to the k pure quantum states of the second Hamiltonian projected onto each of multiple eigensubspaces of the first Hamiltonian.

[034] The Hilbert space has a dimensionality of 2^{N} and the subspace has a dimensionality of IM , where k and / and M are integers and / < k .

[035] A quantum state representing the quantum computer and subject to the total Hamiltonian undergoes adiabatic dynamics that are confined to the subspace.

[036] In another aspect, in general, a quantum computing system comprises a scheduling computer configured to control a quantum computer by performing all the steps of any of the methods.

[037] In another aspect, in general, software comprises instructions stored on a non- transitory computer-readable medium for causing a scheduling computer to control a quantum computer by performing all the steps of any of the methods.

[038] Aspects can have one or more of the following advantages.

[039] The techniques described herein are able to identify either optimal or sub- optimal schedules that are better than both the linear schedule and the problem- independent schedule, providing superior performance for AQC in many applications.

[040] The (sub-)optimal schedules are devised by analyzing the spectral gap of the AQC quantum Hamiltonian.

[041] The calculation of the spectral gap is impossible without reducing the dimensionality of the AQC Hamiltonian for the desired problem. Therefore, we provide techniques to effectively and efficiently reduce the dimensionality of AQC Hamiltonians.

[042] Unlike previous techniques which are based on the existence of explicit symmetries of the AQC Hamiltonian, the techniques described herein are not limited to these particular classes of problems. Instead, the effective dimensionality can be reduced without relying on any explicit symmetry. The techniques also go beyond the strict distinction of driver and problem contributions to the Hamiltonian, which is made possible by the recognition of a hidden block diagonal structure in the total Hamiltonian that leads to the identification of a small subspace in which the relevant part of the energy spectrum, and in particular the energy gap between ground state and first excited state, can be efficiently computed. The relevant eigenstates are effectively confined to this small subspace. Using this more efficient representation, a scheduling computer can be configured to efficiently and exactly compute, at any point during the adiabatic process and at zero temperature, the energy spectrum of an effective Hamiltonian. The computation can be performed using quantities that can be computed exactly or using efficient numerical approaches that require only a polynomial amount of classical resources.

[043] Other features and advantages of the invention are apparent from the following description, and from the claims.

Description of Drawings

[044] FIG. 1 is a block diagram of a quantum computing system.

Description

[045] A strategy for determining a schedule that is closer to optimal than a simple linear schedule is based on additional characteristics of the specific problem, or class of problems, to be solved. In some implementations, the strategy includes choosing s(t) to be the solution of the following differential equation

with the energy gap between the ground state and the first excited state of H[046] In both the linear case and the nonlinear case described above, the spectrum of H_{AQC} is needed to compute the schedule for controlling the quantum computer.

[047] In one aspect, the calculation of the spectrum of an arbitrary H _{AQC} is accomplished by identifying a convenient splitting of this total Hamiltonian into two terms representing distinct contributions that simplify at least some of the subsequent operations. The two terms are defined using the two Hamiltonians H_{A} (s) and H_{B} (s)

(with time dependence of s being implicit), where their sum with appropriate coefficients is able to reproduce the total Hamiltonian H _{AQC} (t) , corresponding to each value of s{t) , as follows

[048] The choice of H_{A} (s ) and is made according to the following criteria.

[049]

H is a highly degenerate Hamiltonian with only M distinct energy levels. is the sum of k projectors on pure quantum states (also called "rank-1"projectors, or "single-state" projectors, or "unidimensional" projectors).

[050] It should be understood that the techniques described in this document are not limited to the particular case in which H_{A} (s) corresponds to either H_{p} or H_{D} . The spectrum of the total Hamiltonian H _{AQC} (t) is calculated by using information extracted from the Hamiltonians H_{A} (s ) and H_{B} (s ) , as described in more detail below.

[051] Under some quite general conditions, the quantity of classical computational resources used by these techniques to compute the spectrum of H_{AQC} grows only polynomially with N as opposed to direct methods involving the solution of eigenvalue problems for matrices with dimensions

100, a problem specification 102, including data representing a problem Hamiltonian and a driver Hamiltonian are provided to a (Classical) scheduling computer 104. The scheduling computer 104 includes at least one processor 106 that executes a computer program 108 for generating the schedule data 1 10 that will be used to control a quantum computer 1 12. The scheduling computer 104 executes the program 108 to apply an analysis procedure to determine the minimum spectral gap for the input

Hamiltonians, and use this minimum spectral gap to determine schedule data (e.g., including a representation of the schedule function s(t) at multiple times between a start time of t = 0 and an end time equal to the a total evolution time T ) that is passed to the quantum computer 1 12. A controller 1 14 in the quantum computer 1 12 then controls signals coupled to an arrangement of qubits 1 16 of the quantum computer 1 12 according to the schedule data 1 10, and ultimately provides a problem solution 1 18 according to a final configuration of the arrangement of qubits 1 16 of the quantum computer 1 12 after operation for the time T .

[053] In embodiments in which the quantum computer 1 12 is an adiabatic quantum computer, operation of the quantum computing system 100 includes selecting an evolution time T , and then operating the quantum computer during an interval of time between 0 and T according to an evolution schedule function s{t). Such a schedule function represents the time-variation of the Hamiltonian that governs the dynamics of the quantum computer 1 12. The change of the Hamiltonian is given by a time-variation of the degree of combination of a first Hamiltonian representing a default initial choice (e.g., a "driver" Hamiltonian) and a second Hamiltonian encoding the problem to be solved. While the evolution schedule may be determined based on data representations of the first and the second Hamiltonians, in some embodiments, the computations performed by the scheduling computer 104 include determining the evolution schedule based on data representations of modified Hamiltonians H_{A} (s) and H_{B} (s ) , which enable additional efficiency in the Classical computation, as described in more detail below.

[054] When the scheduling computer 104 has been configured with the total Hamiltonian H _{AQC} (t) divided into the distinct contributions H_{A} (s ) and H_{B} (s ) , a sequence of computations are performed to obtain a basis in which the total Hamiltonian results in a block diagonal matrix for every s. The data representation of
the only relevant block is obtained by selecting the only block whose size is larger than l x l .

[055] While the specific equations used to perform these computations are described in greater detail below, the following is a high-level summary of the steps involved in the schedule generation program 108. A first step is projecting each of the k states involved in H_{B} (s) on the eigensubspaces of H_{A} (s) . This results in kM states, labelled by { | E_{a}) } , so that those states belonging to different eigensubspaces of H_{A} (s) are orthogonal, while the k states belonging to the same eigensubspace are not necessarily orthogonal. This calculation involves the normalization of the projected states.

[056] A second step includes calculation of an overlap matrix, whose entries are expressed below. The overlap matrix has dimensions (k M) x (k M) .

[057] The normalization of the \ E_{a} } states and the construction of their overlap matrix make use of a preliminary representation of the Hamiltonian H _{A} (s) , in particular of the structure of its eigensubspaces. In several cases of interest (for example when H_{B} (s) corresponds to a Graver style driver Hamiltonian and H_{A} (s) is a generic problem matrix, a representation of the density of states of H_{A} (s) is sufficient. For more general situations, multi-point correlation functions at fixed energy have to be provided in addition to the density of states.

[058] Preliminary input representation of H _{A} (s) is available in at least the following three different ways: analytical analysis of the problem that gives its "a priori" characterization, Monte Carlo simulations in the form of entropic sampling, or possible future methods and data algorithms to evaluate the necessary properties.

[059] A third step includes extracting from the \ E_{a}) states a smaller set of linearly independent, pairwise orthogonal states \ ε_{α})■ This is achieved by applying the

Cholesky decomposition to the overlap matrix. Since the overlap matrix is a Hermitian positive-semidefmite matrix, its Cholesky decomposition may be not unique, but it always exists. This reflects a certain freedom in choosing the states \ ε_{α}) , but their number is always the same and never larger than k M .
[060] A fourth step to obtain the desired basis is to complete the set { \ ε_{α}} } into a basis of the complete space. This can be done by adding pairwise orthogonal states belonging to the eigensubspaces of H_{A} (s) and orthogonal to all \ ε_{α}) .

[061] The only subspace relevant to AQC is the one spanned by the states \ ε_{α}) . When the total Hamiltonian H _{AQC} (t) is represented in the basis constructed in the four steps above, this relevant subspace appears as a separate block along the diagonal of the resulting block diagonal matrix. In fact, this block is the only block with dimension larger than 1 x 1 .

[062] The relevant diagonal block of the total Hamiltonian H _{AQC} (t) is now represented by a matrix of dimensions smaller than (k M) x (k M) , where k is the number of states on which the Hamiltonian H_{B} (s) acts non-trivially and M is the number of distinct energies composing H _{A} (s) . While in the most generic case (kM^ can be exponential in N, if we consider H_{A} (s) to be the problem Hamiltonian corresponding to many concrete cases (3 SAT, QUBO, Max-2SAT, Exact Cover, . ..), then M is polynomial (quadratic or cubic) in N. In this case, the value of k depends on the explicit choice of the driver Hamiltonian. The consideration above is not intended to limit the applicability of the techniques described herein to the particular case H_{A} (s) = H_{p} , but to provide a concrete example where M is polynomial, and typically only quadratic or cubic, in N.

[063] The Hamiltonian restricted to the subspace generated by the states \ ε_{α}) is called the "effective" Hamiltonian. The relevant part of the energy spectrum of the total Hamiltonian H _{AQC} (t) is obtained by solving the eigenvalue problem associated with the effective Hamiltonian at any instant (in practice every small step As ) during the adiabatic evolution.

[064] The reduction method introduced herein allows the calculation of the energies of H_{AQC} (s) to be performed on an ordinary classical computer with order of (kM)^{3} operations. Any algorithm devised to diagonalize matrices can be used to such aim.

[065] The difference between the two lowest eigenenergies determines the spectral g Oap I . Its minimum value between s = 0 and s = 1 is denoted g o mm. . From g o mm. it is possible to estimate the evolution time expected for a successful computation as

[066] The knowledge of the spectral gap g(s) , obtained thanks to the reduction method, leads to the optimal schedule as described above. Such optimal schedule can improve the scaling of T on the problem size N .[067] Some AQC or quantum annealing machines operate according to a fixed, problem independent, schedule. The techniques described herein provide a schedule that is closer to an optimal schedule, tailored either on the specific problem of interest or on its specific class of problems.

[068] These techniques concern not only the calculation of the spectral gap g(s) , but of the full spectrum (eigenvalues and eigenstates) of the relevant part of the total Hamiltonian.

[069] The following description includes examples of equations that can be used to program the scheduling computer 104.

[070] The driver Hamiltonian H_{D} can assume a variety of forms, but only a few regularly appear in the literature: The "Grover style" driver Hamiltonian (or simply Grover driver Hamiltonian),

with corresponding to the equal superposition of all the states | JC) ,

and the "standard" driver

where is the X Pauli matrix acting on the i -th qubit, which physically corresponds to a quantum transverse field. Despite their diversity, both H_{D} are invariant under the exchange of any pair of qubits, have the same ground state and do not commute

with any H_{p} apart from the trivial H_{p}∞ 1 case.

[071] To determine whether the system evolution is adiabatic, several conditions have been proposed that all relate the evolution time T to the inverse of the spectral gap

, where E(s) and E ^{'} (s) are, respectively, the ground state energy

and the first excited state energy of the adiabatic quantum Hamiltonian H_{AQC} (t^ . In
particular, a widely adopted condition implies with the

minimum spectral gap achieved during the evolution from t = 0 to t = T .

[072] As mentioned above, and do not necessarily correspond to the

initial driver or problem Hamiltonian and, in general, depend on s . In the following, we will omit any obvious dependence on s to maintain the notation as readable as possible.

[073] A main idea is that, among the many possible choices of H_{A} and H_{B} , the procedure performed by the scheduling computer 104 searches for those in which H_{A} is a highly degenerate Hamiltonian with only M distinct energy levels and H_{B} consists of the sum of k rank-1 projectors, namely

with

and pairwise orthonormal states. Ωwith the eigenvalue E of H_{A} and Ρ_{Ω} the corresponding projector. Therefore, the

E

procedure will lead to an exponential reduction whenever both k and M depend polynomially on n .

[074] The non-trivial action of H _{AQC} (t) is limited to the space spanned by the set of states | E_{a}) oc Ρ_{Ω}^ \ ψ_{α})■ Restricting the action of the Hamiltonian to this subspace, we obtain

where is a normalization factor and the states ^ are

given by the orthogonalization of the set . Here x(E) < k is the actual

number of linearly independent at given energy E . As a consequence, the

Hamiltonian H_{eS} (s) results to be an effective (K x M) -level Hamiltonian, where

K We want to emphasize that the effective Hamiltonian is not an

approximated version of the original H_{AQC} (t) , but an exact description of its relevant
part. In fact, if we extend the set to a complete basis by adding orthonormal

vectors belonging to eigensubspaces of H_{A} , then H_{AQC} (i) presents a peculiar block diagonal structure when represented in such basis: the only block with dimension larger than l x l is a (KM) x (KM) block exactly reproduced by

[075] It is important to appreciate a subtlety: in most cases, we do not know the exact form of the states | E_{a}) , for example because they are related to the eigenstates of H _{A} .

Then, how can we obtain the explicit entries of to configure the scheduling

computer 104 to perform the numerical analysis? The answer, as described in more detail below, is that all the entries of _{ff} can be computed as far as the overlaps

( at a given energy E are known. For many relevant cases, such overlaps can

be computed either analytically or numerically by using algorithms that require only a polynomial amount of classical resources. Problems with a variety of forms of problem and driver Hamiltonians can be soved. In some cases, the procedure can successfully represent problems in which neither the problem nor the driver Hamiltonian are of Grover-style form.

[076] As described above, the procedure starts with the decomposition of the total Hamiltonian given as the sum of two contributions, H_{A} and H_{B} , having the form: )

In particular, the procedure selects a highly degenerate H_{A} with only M distinct energies, and a Hamiltonian H_{B} formed by a small number k of rank-1 projectors. The relevant part of the energy spectrum of H_{AQC} (t) can be obtained assuming an effective system. Here, we present the derivation in the case of k = 1 , i.e., for a Grover- style Hamiltonian _{>} where the procedure is more intuitive.

[077] In the definition of H_{A} , E represents one of the M distinct eigenvalues and Ρ_{Ω}

E

the associated eigensubspace whose degeneracy is denoted by λ(Ε) . From the completeness of H_{A} we have and For each energy E , we

define as the normalized projection of \ ψ_{α} ) on the subspace P_{n} , and

introduce [ orthonormal states to obtain a basis of

We have

and then

[078] Notice that, while can be non-zero, and are always

orthogonal because

and then

[079] We observe that Eq. (1) describes a Hamiltonian that is block diagonal in the basis

since the terms (5) are factorized from the remaining terms (3) and (4). Thus, the relevant part of the AQC Hamiltonian results in

which is an effective M-level Hamiltonian, where M is the number of distinct energy levels of the contribution H_{A} .

[080] Next, we extend the derivation above to the general form of H_{B} . We will show that it is then possible to reduce a generic AQC Hamiltonian to a (M x K) -level effective Hamiltonian, where M is the number of energy levels of H_{A} and K is an integer number smaller than or equal to the number k of states over which the term H_{B} acts non-trivially. Let us consider the Hamiltonian

where χ are k arbitrary orthogonal states. With a straightforward

generalization of the notation, we introduce

with and divide the subset Ω_{£} in two parts, one spanned by

{| E_{a} )}_{a=l k} and the other representing its orthogonal complement ω_{Ε} . As for the 1 - state case, the set ω_{Ε} is by construction contained in the kernel of H_{B} , such that

for any and for any energy E . As a consequence, all states in ω_{Ε} can be

neglected in the effective Hamiltonian. Moreover, since it is not said that

we use an orthogonalization procedure to extract from the original set a

smaller set of rc(E) < mm{k ,λ(Ε)} pair-wise orthonormal states

[081] In this way

and recalling that

the (relevant part of the) AQC Hamiltonian in Eq. (1) beco

[082] In Eq. (16), we already removed all terms in ω_{Ε} because they are factorized with respect to the relevant part of the AQC Hamiltonian. As one can see, Eq. (16) describes an effective (M x K) -level Hamiltonian, where ■ Correctly, if k = \ we

obtain the AQC Hamiltonian reported in Eq. (10).

[083] It is important to observe that we reduced the original AQC Hamiltonian in Eq. (1) to an effective (M x K) - level Hamiltonian, and then we reduced the Hilbert space from 2" states to (M x K) states. Therefore, if both K and M are polynomial in the number of spins n , the reduced effective Hamiltonian in Eq. (16) can be expressed using
only a polynomial number of states. That is to say that we obtained an exponential reduction of the Hilbert space. We observe that the calculation of Z_{a} (E) and

[084] The states can be expanded as a linear combination of the orthonormal

states which, we recall, span the effective subspace containing the relevant part of

the total energy spectrum. Introducing the rc{E) x k matrix T with entries

Then, we have:

and interpret the above values as the entries of a certain matrix V . Such matrix is a square matrix with linear dimension k and can be shown to be Hermitian and positive- semidefinite. Therefore it admits a Cholesky decomposition:

where U is an upper triangular matrix with real and positive diagonal entries. While every Hermitian positive-definite matrix has a unique Cholesky decomposition, this does not need to be the case for Hermitian positive-semidefinite matrices and this reflects a certain freedom in choosing the states · It appears clear that the expansion

coefficients Τ_{μα} are the entries of a particular choice of such matrix U = T .

[085] Expressin the Hamiltonian H_{B} in the basis of the effective subspace, we have

whereas the term H_{A} becomes

In this way, we have expressed all the necessary operators in the reduced basis. [086] These techniques concern not only the calculation of spectral gap for a total Hamiltonian composed of contributions from only the driver and problem Hamiltonians, but include cases in which a third contribution is considered. This additional term can be associated with noise contributions (see section 4 of Mandra et al., arXiv: 1407.8183v2 [quant-ph], incorporated herein by reference), with a heuristic modification of the total Hamiltonian (see section "Energy gap for multi-parameter family of Hamiltonians" of Mandra et al., arXiv: 1407.8183vl [quant-ph], incorporated herein by reference), with control terms (for example to counteract the anti-adiabatic part of the dynamics), or with other future improvements of the AQC approach. It is to be understood that the foregoing description is intended to illustrate and not to limit the scope of the invention, which is defined by the scope of the appended claims. Other embodiments are within the scope of the following claims.

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US20110047201 * | 3 Aug 2010 | 24 Feb 2011 | Macready William G | Systems, methods and apparatus for factoring numbers |

US20110313741 * | 21 Jun 2011 | 22 Dec 2011 | Spectral Associates, Llc | Methodology and its computational implementation for quantitative first-principles quantum-mechanical predictions of the structures and properties of matter |

Classifications

International Classification | G06F17/11, G06F17/10 |

Cooperative Classification | G06N99/002 |

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