Quantum solver

Abstract

This disclosure relates to a method for determining a characteristic of a wave function of electrons. The method comprises running a quantum circuit multiple times. The quantum circuit comprises qubits and is configured to represent an eigenstate of a Hamiltonian of the electrons. After each time of running the quantum circuit, a value is measured of each qubit to obtain multiple bit strings encoding respective basis functions. The method determines measurement statistics over the bit strings and calculates, based on the measurement statistics a wave function coefficient, including a sign value of each measurement, for each basis function encoded by the bit strings. Then, the method calculates an energy value of the wave function based on the wave function coefficient for each basis function and repeats these steps adjusting parameters of the quantum circuit to iteratively minimise the energy value to thereby improve the characteristic of the wave function.

Description

"Quantum Solver"Technical Field

[0001] This disclosure relates to a quantum solver and in particular, to a quantum solver to determine the wave function of electrons.Background

[0002] Quantum computers aim to provide a quantum advantage over classical computers in the near future. One promising area is the determination of wave functions of electrons of atoms or molecules. This may assist the exploration of a large number of different molecules by the quantum computer without time consuming physical experiments.

[0003] However, approaches like the variational quantum eigensolver have not yet provided a quantum advantage, one main reason being that the number of measurements becomes too large for practical applications.

[0004] Any discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present disclosure as it existed before the priority date of each of the appended claims.

[0005] Throughout this specification the word "comprise", or variations such as "comprises" or "comprising", will be understood to imply the inclusion of a stated element, integer or step, or group of elements, integers or steps, but not the exclusion of any other element, integer or step, or group of elements, integers or steps.Summary

[0006] This disclosure provides a quantum solver that samples a wave function and calculates wave function coefficients. The disclosed quantum solver is (i) faster, (ii) more accurate, and / or (iii) less energy demanding over the classical competitors.

[0007] A method for determining a characteristic of a wave function of electrons comprises:(i) running a quantum circuit multiple times, the quantum circuit comprising multiple qubits and being configured to represent an eigenstate of a Hamiltonian of the electrons;(ii) after each time of running the quantum circuit, measuring a value of each of the multiple qubits to obtain multiple bit strings encoding respective basis functions;(iii) determining measurement statistics over the multiple bit strings;(iv) calculating, based on the measurement statistics and using a classical or an additional quantum computer a wave function coefficient, including a sign value of each measurement, for each basis function encoded by the multiple bit strings;(v) calculating an energy value of the wave function based on the wave function coefficient for each basis function;(vi) repeating steps (i)-(v) and adjusting parameters of the quantum circuit to iteratively minimise the energy value to thereby improve the characteristic of the wave function.

[0008] It is an advantage that the method directly determines the wave function coefficients for each of the multiple bit strings because fewer measurements are required compared to repeatedly measuring Pauli strings and adjusting the quantum circuit to iteratively minimise the ground state energy. Therefore, the method can provide near-term quantum utility, which means that the method can outperform classical solvers with a similar physical footprint, especially because the method can be implemented in parallel in many quantum circuits. In some embodiments, the step (iv) of calculating the wave function coefficient comprises calculating, using the classicalor the additional quantum computer, sign values for the measurements and adjusting the measurement statistics based on the sign values.

[0009] In some embodiments, calculating the sign values comprises calculating a sign value for each of the multiple bit strings.

[0010] In some embodiments, calculating the sign value is based on multiple weights; and the method comprises before step (vii) repeating steps (iv)-(vi) and adjusting the multiple weights using a classical or an additional quantum computer to iteratively minimise the energy value.

[0011] In some embodiments, the characteristic comprises the wave function coefficient for each basis function or a binding energy provided by electrons.

[0012] In some embodiments, the method comprises calculating an initial value of the multiple qubits based on mean field electron interactions.

[0013] In some embodiments, each bit string encodes one Slater determinant.

[0014] In some embodiments, the method further comprises filtering Slater determinants by keeping only Slater determinants with a minimum number of |1> values.

[0015] In some embodiments, each bit string encodes a configuration state function represented by a linear combination of Slater determinants.

[0016] In some embodiments, each of the multiple bit strings is associated with an occupation number vector.

[0017] In some embodiments, the method comprises: performing, in the quantum circuit parameterised single qubit gates and controlled multi-qubit gates; and adjusting parameters of the quantum circuit comprises adjusting the parameters of the single qubit gates.

[0018] In some embodiments, running the quantum circuit comprises operating qubits integrated into a diamond substrate.

[0019] In some embodiments, the method further comprises broadening the measurement statistics.

[0020] In some embodiments, broadening the measurement statistics comprises adding counts for bit strings that are in an equivalence class of a measured bit string.

[0021] A method implemented on a classical computer to determine a characteristic of a wave function of electrons comprises:(i) for each of multiple runs of a quantum circuit, receiving a measured value of each of multiple qubits of the quantum circuit, to obtain multiple bit strings encoding respective basis functions;(ii) determining measurement statistics over the multiple bit strings;(iii) calculating , based on the measurement statistics, a wave function coefficient, including a sign value of each measurement, for each basis function encoded by the multiple bit strings; and(v) calculating an energy value of the wave function based on the wave function coefficient for each basis function.

[0022] Software, when executed on a classical computer, causes the classical computer to perform the above method.

[0023] A computer system comprises a classical processor configured to determine a characteristic of a wave function of electrons, by performing the above method.

[0024] A system comprises a quantum circuit, the quantum circuit comprising multiple qubits and being configured to represent an eigenstate of a Hamiltonian of the electrons; and a classical computer configured to perform the above method.Brief Description of Drawings

[0025] Figure 1 illustrates orbitals of a water (H2O) dimer.

[0026] An example will now be described with reference to the following drawings:

[0027] Figure 2 illustrates an architecture 200 for a variational quantum method, according to an embodiment.

[0028] Figure 3 illustrates the architecture from Figure 2 in more detail, according to an embodiment.

[0029] Figure 4 illustrates how the calculated sign value is applied to the measurements, according to an embodiment.

[0030] Figure 5 illustrates a method for determining a characteristic of a wave function of electrons, according to an embodiment.

[0031] Figure 6 illustrates a classical computer system for determining a characteristic of a wave function of electrons, according to an embodiment.

[0032] Figure 7 illustrates an encoding circuit for the disclosed second-quantized quantum wave function sampler approach of the H4 benchmark system initializing a multi-reference state, according to an embodiment.Description of Embodiments

[0033] Electrons are the main factor in most chemical bonds. Therefore, it is important to determine the electronic structure of atoms and molecules in order to predict their bonding behaviour. For example, electrons on the valence band can interact with other electrons in order to create a bond.

[0034] For the simplest atom, the hydrogen atom, the electronic structure is governed by an orbital model that conceptually uses spherical orbitals to represent the different energy levels of the electrons. The spherical orbitals of a single hydrogen atom can be described analytically. For more complex atoms and molecules, the electronic structure becomes more complex. In particular, the electrons are correlated because of the interactions between electrons. This makes the exact calculation of the electronic structure impossible in many cases using existing classical computers.

[0035] Figure 1 illustrates orbitals of a water (H2O) dimer to illustrate the complex nature in the case of a relatively simple molecule. It is noted that each orbital describes one particle state but does not describe the actual many body wave function. In order to determine the wave function, all possible combinations of anti- symmetrised products of orbitals need to be considered. This problem scales badly and is therefore not solvable accurately with current classical computer hardware for more than few (e.g., 10) electrons. Therefore, there is a desire to use quantum circuits to determine the electronic structure. The idea is that the superposition and entanglement between qubits can naturally incorporate correlations between electrons without the scaling problems associated with classical computers.

[0036] Mathematically, the electronic structure is often represented by an electronic wave function and it is the aim of the methods in this field to find an accurate estimate of the electronic wave function. One principle that can be used is the variational principle, which states that the energy expectation value of any trial wave function is an upper bound to the exact energy. An optimal trial wave function is therefore obtained by minimizing their respective energy expectation value.

[0037] Figure 2 illustrates an architecture 200 for a variational quantum method. The architecture 200 comprises a quantum register 201, a quantum circuit 202, a measurement module 203 and a feedback 204. It is noted that the quantum circuit 202 may be a physical circuit including physical qubits or may be an emulated circuit that classically emulates the physical circuit using a classical model including noise characteristics of the actual physical hardware.

[0038] The quantum register 201 is initialised to quantum states to provide a useful starting point for calculating the wave function. In some examples, the quantum states are set according to previous calculations or estimates of the electronic structure. For example, a mean field method is computationally feasible because it does not consider correlations between electrons. The result of the mean field method is a set of orbitals. These can then be mapped to the quantum register. For example, an occupied orbital can be mapped to a |1> qubit state and an unoccupied state can be mapped to a |0> qubit state.

[0039] Quantum circuit 202 comprises a number of operations performed on the qubits in register 201, such as rotations, controlled rotations and other operations. The operations are unitary, which is why the circuit 202 is denoted by “U” in Figure 2. The operations are parameterised, such as by rotation angles, for later adjustment. Quantum circuit 202 may be designed specifically for the molecule in question and to represent an eigenstate of a Hamiltonian of the electrons. In general, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues is the set of possible outcomes obtainable from a measurement of the system's total energy.

[0040] Once the operations of quantum circuit 202 have been carried out, the qubits of register 201 are in a resulting state which can be measured. Measurement module 203 measures those qubits. Each qubit may be in a superposition of its two pure states, for example, |0> and |1>. According to the laws of quantum mechanics, the measurement collapses that superposition into one of the states |0> or |1> for each qubit depending on the degree of superposition. The measurement module 203 then calculates the energy of the system and the feedback 204 adjusts the quantum circuit 202 to iteratively reduce the energy calculated by measurement module 203. Over time, the architecture 200 arrives at a minimum energy, which means the qubits after applying the quantum circuit 202 with optimised parameters, represent the optimal electronic wave function. This enables the calculation of any observable, such as the ground state energy and / or excited states energy values as well as molecular properties.

[0041] In cases where the measurement module 203 measures Pauli strings to essentially measure the Hamiltonian and then derive the energy, it has been found that the number of measurements may not scale well. As a result, it may be difficult to achieve a quantum advantage on current quantum hardware, that is, the quantum hardware may not outperform classical computers in some cases. For these cases, this disclosure provides a measurement module 203 that reduces the number of required measurements significantly.

[0042] More specifically, the architecture 200 runs the quantum circuit 202 multiple times, such as 1,000 times and builds measurement statistics, such as a histogram of measured qubit states. Each measurement of the qubit register 201 is referred to as a bit string because the measurement results in a string of either ‘0’ or ‘ 1’ for each qubit. The measurement module 203 counts how many times each bit string occurred.Further, measurement module 203 maps each bit string to a basis function of the wave function and can calculate, from the histogram counts, coefficients of the basis functions. With those coefficients, the measurement module 203 constructs the wave function or more specifically, calculates the energy of the current solution. The feedback 204 can then adjust the parameters to minimise that energy.

[0043] There is one particularity in the energy calculation in that this calculation may use a sign value for each measurement. While this sign value may not be physically measurable, measurement module 203 can optimise the sign value together with the parameters of quantum circuit 202 to minimise the calculated energy. It has been found that the optimisation of the sign value converges relatively quickly, which means this method provides good performance. Importantly, the processing time of this architecture scales better than when measuring Pauli strings, for example.

[0044] It is noted that the sign value is an example for the phase. So in other examples, other values of the phase can be calculated. That is, measurement module 203 can optimise the phase value together with the parameters of the quantum circuit 202 to minimise the calculated energy as disclosed herein. This is relevant when there are phase dependent terms in the Hamiltonian that require the phase in order tocalculate the energy. In that case, the energy calculation is also based on phasedependent terms.

[0045] Further, in some embodiments, the qubit strings do not contain the qubit values explicitly but may contain calculated values that are based on the qubits. For example, there may be a machine learning method, such as a neural network, that has the qubit values as input and the values of the bit string as an output. The parameters of the machine learning method, such as the neural network, can also be optimised within the optimisation routine disclosed herein to minimise the energy value. As set out above, the measurement module determines measurement statistics oven the multiple bit strings and then calculates, based on the measurement statistics, a wave function coefficient for each basis function encoded by the multiple bit strings. The wave function coefficient includes a sign value of each measurement. This can be done by adjusting the measurement statistics based on the sign value or by calculating the coefficient in another way that implicitly includes the sign value.

[0046] In yet a further embodiment, the wave function is not an electronic wave function but that of other fermions or other particles. The variational principle also holds for those cases, which means the disclosed method can be applied.

[0047] Figure 3 illustrates architecture 200 in more detail comprising, again, the quantum register 201, quantum circuit 202 and measurement module 203. It can be seen that each measurement z results in a bit string |bi>. It is noted that more accurately, the measurement here is referred to as a “shot” because it measures the outcome of a single run of quantum circuit 202. So each bit stringis for a shot z and qubits 0-3. Each bit value in the bit string is either ‘0’ or ‘ 1 ’ . There are also shown weights w with one weight for each bit value. The weights w are weights of a weighted sum of bit values and the result, weighted again by a single weight, is the sign value of each shot (i.e. each sampled bit string). It is noted that these calculations are performed on a classical computer. Since the correlations between electrons are inherently represented by superposition states in the quantum circuit, the classical computation scales well with problem size.

[0048] It is noted that the weights are not per shot but only per qubit. That is, the weights have the same value for all 1,000 shots, for example. It will be explained below how the weights are calculated.

[0049] Figure 4 illustrates how the calculated sign value is applied to the measurements. Since the weights are the same for all shots, the sign is not applied to each shot individually but can be applied later to the probability in the histogram. That is, the measurement module 203 counts the occurrence of each bit string, which, once normalised, results in the probability distribution 401 in Figure 4. Then, the measurement module 203 calculates for each bit string the sign value by multiplying each weight with sign=+l for bit value ‘0’ and sign=-l for bit value ‘ 1’. Measurement module 203 then applies the sign value to that probability. This results in the wave function coefficients 402 also shown in Figure 4. Alternatively, it is also possible to apply the sign value to each shot measurement. It can be seen that for the bit string 0101, the sign is negative, which means the coefficient for that basis functions is shown negative in 402. It is further possible that the sign value is included into the wave function coefficient in other ways.

[0050] The basis functions can be Slater determinants, linear combinations of Slater determinants or other basis functions. So, in order to calculate the energy resulting from the calculated coefficients, measurement module 203 uses the mapping from energy states to qubits in reverse to calculate the wave function or the resulting energy. The measurement module 203 can now adjust parameters to reduce that energy. Interestingly, the measurement module 203 can, with purely classical calculations, adjust the weights w to minimise the resulting energy. That is, measurement module 203 changes each w individually and then follows the steepest gradient. This process is similar to the training of a neural network, which is why it can be said that the sign value is calculated by a single-layer neural network with edge weights w. It is noted, however, that different structures, such as multiple layers and different classical or even quantum machine learning methods could equally be used.

[0051] Once the w weights are optimised, it is possible to run another round of 1,000 shots to also optimise the parameters of the quantum circuit 202. Once the energy cannot be further reduced, or the results do not change or a maximum number of iterations has been reached, the resulting coefficients can be considered to accurately represent the desired wave function.

[0052] Figure 5 illustrates a method 500 for determining a characteristic of a wave function of electrons, such as coefficients of basis functions or binding energy provided by the electrons. The method commences by running 501 a quantum circuit multiple times. As described above, the quantum circuit comprises multiple qubits and is configured to represent an eigenstate of a Hamiltonian of the electrons. The multiple qubits are initialised (also referred to as “input qubits” to represent occupation vectors) according to a mapping from the physical system to the qubit space. This mapping to qubits is achieved by methods referred to as “second quantized encoding” and the actual values can be calculated by mean field electron interactions, for example. Other mappings, such as a qubit-efficient mapping (see below) or others, may equally be used.

[0053] In one example, the encoding may comprise a ternary tree encoding for ab initio molecular systems or generalized superfast encoding for lattice models. This results in 1 qubit per 1 orbital. More information on setting up the encoding and the quantum circuit can be found in Tilly, Jules, et al. "The variational quantum eigensolver: a review of methods and best practices." Physics Reports 986 (2022): 1- 128, which is incorporated herein in full by reference.

[0054] Method 500 further comprises, after each time of running the quantum circuit (e.g. each shot), measuring 502 a value of each of the multiple qubits to obtain multiple bit strings. This means one bit string is sampled at each shot and each bit string sample is one of a limited number of possible bit strings. Since there is little source of confusion “bit string samples” and “bit strings” are used interchangeable herein as it should be clear from context. Each bit string encodes a respective basis function. For example, bit string “0101”, which may be sampled about 800 times out of 1,000 totalsamples, represents a dominant basis function (as shown at 401 in Figure 4). In one example, each bit string encodes exactly one Slater determinant, while in other examples, each bit string encodes a linear combination of Slater determinants. A Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions). For an N- electron system, the Slater determinant may be defined aswhere the last two expressions use a shorthand for Slater determinants: The normalization constant is implied by noting the number N, and only the one -particle wave functions (first shorthand) or the indices for the orbital labels (second shorthand) are written down. All skipped labels are implied to behave in ascending sequence.

[0055] In some examples, for improving performance, there may be a filter that only accepts bit strings with a minimal number of Is, so to avoid empty bit strings that encode solutions with insufficiently occupied orbitals, which do not conserve the total particle number and therefore do not reflect the physical system.

[0056] Method 500 then determines 503 measurement statistics over the multiple bit strings, such as a histogram of counts of occurrences of each bit string in the samples, which may be normalised. As described above, using weights w, method 500 calculates 504, using a classical computer, a sign value for each of the bit strings, such as by calculating a weighted sum and applying a tanh function to map the result to -1 and 1.

[0057] Once the sign value is available, method 500 adjusts 505 the measurement statistics based on the sign value to determine a wave function coefficient for each basis function encoded by the multiple bit strings and then calculates 506 an energy value of the wave function based on the wave function coefficient for each basis function. This way, the sign value is included into the wave function coefficient. At this point, method 500 may optimise weights w classically for the sign calculation to reduce the energy without running the quantum circuit again. However, the sign value may also be included into the wave function coefficient in other ways. Once finished, method 500 repeats steps 501-506 and in each repetition adjusts the parameters of the quantum circuit to iteratively minimise the energy value. This means, the method runs another 1,000 shots (for example) to obtain measurement statistics, sign value, coefficients and energy value. By repeating these steps while adjusting the circuit parameters, method 500 improves the characteristic of the wave function.

[0058] In one example, the method 500 may further comprise the step of broadening the measurement statistics, which is also referred to as histogram broadening. The idea of histogram broadening is to augment the collected measurement statistics of the quantum wave function sampler by an empirical correction based on the originally measured bit string and its number of counts. More specifically, for any sampled bit string, method 500 forms an equivalent class. The equivalent class is a set of bit strings that are logically connected to the sampled bit string. In one example, the logical connection is a logical operation, such as a bit left shift for example, and each bit string that results from applying that logical operation to the sampled bit string one or more times is included into the equivalent class. For each string in the equivalence class, method 500 assigns a number of additional counts. This number may be decreasing based on how many times the logical operation has been applied to result in the current bit string. The decrease may be calculated by an exponential or linear or other decrease.

[0059] In one illustrative example, bit string “0110” has 1,000 measurement counts. This bit string can be shifted left multiple times to create the following bit stings in the equivalent class: 0110, 1100, 1001, 0011. After the last element, the left shift results in the original bit string, so the shifting stops. Now the method 500 applies a decreasingadditional count to 1100, 1001 and 0011. The final histogram of this equivalent class may look as follows - noting that counts from other bit strings may be added:0110 ################################################## 1100 ######################### 1001 ##########0011 ##

[0060] Therefore, the collected measurements of "0110" gives rise to additional counts for three other (not directly sampled) bit strings. This may lead to an improved solution of the quantum wave function sampler by generating more bit strings than originally found with potentially very low probabilities (which add up to a nonsignificant ground state energy contribution for larger molecules).

[0061] It is noted that the method directly determines the wave function coefficients including respective sign values for each of the multiple bit strings. Therefore, fewer measurements are required compared to repeatedly measuring Pauli strings and adjusting the quantum circuit to iteratively minimise the energy. Therefore, method 500 can provide a near-term quantum advantage.

[0062] It is further noted that the disclosed method is particularly useful when implemented in quantum computers with diamond qubits (nitrogen vacancies). Those diamond qubits are operational at room temperature and therefore do not require sophisticated cooling apparatuses. As a result, diamond qubits can be deployed in a similar way to ordinary computers, which is in contrast to complex supercomputers. This means, a quantum advantage can be achieved once the quantum solver achieves an advantage over an ordinary computer with a similar resource overhead to the quantum computer.

[0063] Further, the disclosed method lends itself to parallelization by multiple quantum computers. This means that if multiple (such as 100) diamond quantum computers are installed at the same site, a classical computer can control the circuitparameters in each quantum computer and so the different circuit parameters can be evaluated in parallel in the multiple quantum computers. Also, the individual measurement shots can be distributed to all parallel quantum devices (e.g., 100 shots on 10 quantum computers yield 1000 measurements roughly 10 times faster than a single quantum computer). This reduces the optimisation time drastically because the gradient can be determined more time-efficiently. This is much harder when achieving an advantage over more resource intensive quantum computers as those are more difficult to install in parallel.

[0064] Figure 6 illustrates a classical computer system 600 for determining a characteristic of a wave function of electrons. “Classical” in this context means that computer system 600 is not a quantum computer and therefore does not use the concepts of superposition and entanglement of qubits. Instead, a classical computer system comprises a processor 601 that uses voltage levels in a mostly binary manner with transistors to implement logic gates. Examples are current AMD Ryzen or Intel Core processors. Processor 601 may be integrated with a quantum processing unit (QPU) comprising the quantum circuit and the qubits into a single device, such as on a single chip or interconnected on a single board. The QPU may be a diamond-based QPU manufactured by Quantum Brilliance, Australia.

[0065] Processor 601 is connected to program memory 602 and data memory 603. Program memory 602 is a non-volatile computer readable medium with program code stored thereon. When processor 601 executes the program code stored on program memory 602, this causes processor 601 to perform the methods disclosed herein.

[0066] In particular, processor 601 receives for each of multiple runs of a quantum circuit, a measured value of each of multiple qubits of the quantum circuit. This way, the processor 601 obtains multiple bit strings that encode respective basis functions. It is noted that the computer system 600 may be remote from the quantum computer (not shown in Figure 6). That is, the quantum computer may be located in a different facility anywhere on the world and the computer system 600 receives the measurements over the Internet, for example. Processor 601 is also able to remotely send commands to thequantum computer or configure parameters of the quantum computer as explained herein.

[0067] In another example, computer system 600 and the quantum computer are integrated into a single device. For example, the quantum computer may be an extension card or extension device, such as a room temperature diamond quantum computer that is directly connected to the computer system 600. This has the advantage of faster communications, less overhead, more compact architecture and improved data privacy because no external communication (e.g. to / from the cloud) is necessary.

[0068] Once the measurements are received, processor 601 determines measurement statistics over the multiple bit strings and calculates wave function coefficients including sign values for the measurements as explained herein. In some examples, the processor 601 adjusts the measurement statistics based on the sign value to determine a wave function coefficient including the sign value for each basis function encoded by the multiple bit strings and finally calculates an energy value of the wave function based on the wave function coefficient for each basis function. The energy value characterises the wave function. It is now possible for the processor 601 to minimise this energy by adjusting the parameters of the quantum computer.

[0069] More particularly, processor 601 aims to solve the time-independent Schrodinger equation (also referred to as electronic many-body problem in quantum chemistry),by means of minimizing the Hamiltonian expectation value,for an approximate trial wave function | ) «| ) . In general, the wave function | ) can be represented as a state vector in the N -particle sector of the Fock space, Fv, viawhere Svdenotes the symmetric group of order N , p(P) the parity of the permutationthe j -th one-particle function, i.e., spin orbital. The state vectors | i) are generally called Slater determinants. The major task of the disclosed methods for accurate correlation (also called post-Hartree-Fock) is to find an optimal set of coefficients {c;} . For N particles in a fixed set of M one-particle functions, the number of possible Slater determinants increases factorially with such thatexploring the full space Fvis only possible for very small molecular or atomic systems. In contrast, quantum computers can efficiently traverse an exponentially increasing search space with only a linearly increasing number of qubits. Hence, they are promising candidates in the pursuit of more accurate quantum chemical observables.

[0070] As explained above, one encoding to map quantum chemical problems to a qubit register is the second-quantized encoding. In general, each Slater Determinant can be represented by a signed occupation number vector (ONV)where stG {±1 } , and O, V are denoting occupied and virtual orbital spaces. Each ONV encapsulates the full orbital composition of state vector | i) . Please note that this representation neglects the redundant particle orbital mapping as demanded byquantum mechanic’s indistinguishability requirement of particles with overlapping density but lacks the necessary phase information constituted by the order of particle creation in the second quantized picture (see, e.g., a}a |> = + 110010..

[0071] Since the ONVs | btare just regular bit strings of length O u V , they represent a native anchor point for the qubit encoding. Here, each orbital is mapped to an individual qubit such that states 10) and 11) represent empty (virtual) and occupied orbitals, respectively. Through an ansatz circuit, a quantum computer may then utilize superposition and entanglement to construct an ideal ONV representation of the wave function by means of the corresponding bit strings of Hamming weight N (to conserve the total particle number of the investigated system).

[0072] A possible ansatz can be constructed by k products of the exponential of pair coupled-cluster double excitation operators (pCCD), together with generalized single excitation operators (k-UpCCGSD), which has been described in Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley. Generalized unitary coupled cluster wave functions for quantum computation. J. Chem. Theory Comput., 15:311, 2019, which is included herein in full by reference. Other ansatz approaches are also possible, including UCC, such as UCCSD, UCCGSD, symmetry conserving ansatz, hardware-efficient ansatz and others.

[0073] While individual Slater Determinants (and therefore signed ONVs) can conserve the z projection of the spin, Sz, the same may not be true for the total spin, S2. In the non-relativistic case, the molecular Hamiltonian H , commutes with both Szand S2, i.e.,[H, SZ] = [H, S2] = 0, (5)such that any true eigenstate | *+*> ol' / / should be an eigenfunction of both Szand S2. Therefore, the second-quantized or orbital-based encoding may introduce and intermix terms of different spin quantum numbers, which may be referred to as spin contamination. Even if the ansatz circuits are carefully constructed to conserve both Szand S2, quantum noise may still lead to such contamination effects. To circumvent this, a different encoding strategy based on configuration state functions (CSFs) - coined qubit-efficient encoding - may be used. These are special linear combinations of Slater Determinants, which are true eigenfunctions of both Szand S2and may act as the full electronic wave function’s building blocks similar to single Slater Determinants. Further information can be found in: Yu Shee, Pei-Kai Tsai, Cheng-Ein Hong, Hao-Chung Cheng, and Hsi- Sheng Goan “Qubit-efficient encoding scheme for quantum simulations of electronic structure” Phys. Rev. Res., 4:023154, 2022 , which is incorporated herein in full by reference, an example mapping for 2-particle CSFs to bit strings on a qubit register is shown in Table 1.

[0074] Table 1: Mapping example of 2-particle CSFs to bit strings encoded on a 3 qubit register. Overlined indices shall denote a (spin-up) and underlined indices (spin-down) particles.

[0075] With the application of a parameterized ansatz circuit, the quantum device may now proceed as described above with building arbitrary linear combinations of mapped CSFs by superimposing the respective bit strings. In this fashion, any measured probability distribution represents a true S_ and S2eigenfunction. That remains true even in the case of quantum noise.

[0076] More specifically, the quantum wave function sampler (QWFS) disclosed herein samples the wave function itself. This reduces the quantum resources to just a single quantum circuit and may have significant runtime advantages compared to VQEs. The general outline of the QWFS algorithm for a 4-qubit example is illustrated in Figures 3 and 4.

[0077] Following an appropriate qubit encoding of the underlying quantum chemical problem for qubits | q0) to | q3, a parameterized ansatz circuit is applied to the qubit register (step 501 in Figure 5). The latter is used to traverse the qubit’s Hilbert space to find a superposition of bit strings representing the building blocks of the electronic trial wave function | T) . The phase — the sign si— is optimized classically by means of a single neural network-type layer composed of a single node connected to all qubits, i.e., orbitals. For edge and bias weights wk, and w , respectively, the sign siassociated with the bit string | bt) used in step 504 in Figure 5 is given bywhere | bk)) denotes the k -th bit in the bit string | bt) . After execution of the parameterized circuit, bit strings | btmay be measured with a certain probability pt. Together with the classically obtained signs si, these constitute the wave function coefficients via c;= sipi. To update the parameters of the circuit ansatz, the trial energyE is calculated classically. This is one main ingredient in wave function sampling and involves evaluating

[0078] In general, Equation (7) above may scale quadratically with the full number of basis states | i) in the wave function. This amounts to Slater Determinants forN particles in M spatial orbitals in the second-quantized encoding andCSFs for the spin quantum number S in the qubit-efficient encoding.

[0079] Fortunately, in many quantum chemical problems, a majority of these have vanishing or near-vanishing coefficients and need not be sampled and therefore are not included in the energy calculation. This reduces the computational overhead of the classical energy calculation. The quantum device is able to efficiently traverse the Hilbert space and can therefore identify and extract only the important states quickly. Also, there are sufficiently difficult problems in quantum chemistry that use an active space of only a few electrons N . One method to evaluate expressionsinvolves the Slater-Condon rules, which in turn use only the one- and two-particle integrals obtained from a standard mean field calculation (e.g., and usually Hartree - Fock).

[0080] After obtaining the energy (H) , gradient or gradient-free minimization techniques can be used to find an optimal set of ansatz circuit and sign parameters. This involves an iterative procedure implementable on quantum / classical hybridarchitectures of the NISQ era. For the gradient method, the method calculates the gradient in the direction of each of the free variables and then adjusts the parameters by a step size in the direction of the steepest gradient.

[0081] Since the parameterized circuit ansatz does not involve the sign parameters wkand w, the latter may be optimized in a completely classical fashion without requiring additional quantum resources. Intuitively, any changes in w and wkdo not affect the measured probability distribution of bit strings | btbut only the classically calculated energy (H) . The respective gradient of a general parameter p is given bywhere i^(z') = sipidenotes the distribution of wave function coefficients,local energy contribution of bit string | bt) and Dp(i) =the partial parameter derivative. While £’loc(z) occurs in the classical energy estimation (and is therefore evaluated anyway with no extra cost), the derivatives D (z) for the sign parameters w and wkare given by

[0082] More information on this approach can be found in R. Xia and S. Kais. Quantum machine learning for electronic structure calculations. Nat. Commun., 9:4195, 2018, which is included herein in full by reference.

[0083] Clearly, it is computationally efficient to compute Equations (10) and (11) with negligible additional cost to arrive at an analytical gradient for all sign parameters.

[0084] Experiments

[0085] To test the functionality of the QWFS approach disclosed herein, it was applied to several small test molecules as well as the H4 benchmark system of D4h symmetry, which is particularly difficult for classical single-reference methods such as coupled cluster (CC). In Table 2, optimized ground state energies of H2, LiH, and Li obtained from classical exact Full Configuration Interaction (CI) and approximated Complete Active Space CI (CASCI) calculations, as well as the second-quantized and qubit-efficient QWFS methods applied on an ideal state vector simulator and a diamond-based 2 qubit quantum computer, are collected.

[0086] Table 2: Optimized ground state energies in a.u. of the classical exact Full Configuration Interaction (Full CI), Complete Active Space CI (CASCI), and the QWFS methods disclosed herein for ground state energies of the H2, EiH, and Ei2 molecules in the minimal STO-3G basis set. To accommodate all calculations on a 2- qubit quantum computer, the active space approximation was applied to restrict the fermionic excitation space to four CSFs.For the top QWFS row ideal state vector simulations were used with the second- quantized qubit encoding with the linear scaling 1-UpCCGSD circuit ansatz and the L- BFGS-B optimizer employing a convergence threshold of 10’7a.u. The test for the second QWFS row was executed on a 2-qubit diamond-based room-temperature quantum computer integrated at the Pawsey Supercomputing Centre Perth using the qubit-efficient CSF encoding and a hardware-efficient circuit ansatz with an optimization budget of 50 iterations using the ImFil optimizer from skquant (available from https: / / github.com / scikit-quant / scikit-quant and https: / / qiskit.org / documentation / stubs / qiskit.algorithms.optimizers.IMFIL.html). For each iteration, 100 and 250 measurement shots for the Ph and LiH, Li systems, respectively, were used.

[0087] The classically simulated QWFS energies were found in perfect agreement (within the applied convergence threshold) to classical CASCI energies, which are exact in the chosen active space approximation. This indicates that QWFS may successfully sample the exact same wave function as classical methods.

[0088] The same molecular test systems were deployed to a 2-qubit diamond-based room temperature quantum computer integrated at the Pawsey Supercomputing Centre (called the Quantum Development Toolkit (QDK)) using the QWFS approach, a qubitefficient encoding, a hardware-efficient circuit ansatz, and a correction for state preparation and measurement (SPAM) errors. While the Ph energy was obtained within chemical accuracy ( < IO3a.u.) to the exact classical result, the LiH and Lio energies showed deviations of 2.2 10’3and 4.5-103a.u., respectively, compared to the classical exact (within the applied active space approximation) CASCI method. These results show that the physical quantum computer is able to accurately sample electronic wave functions using the methods disclose herein.

[0089] Above, the results are compared to the “ground truth” calculated by Full CI and CASCI. However, those methods are computationally infeasible for larger system. This is why Coupled Cluster (CC) approaches are used for classical simulation of larger molecules. However, the results below show that the classical CC method yieldsinaccurate results for multi-reference (MR) systems, such as the H4 benchmark model where the single Slater Determinant Hartree-Fock approximation is particularly ill- formed. Here, CC methods may not even converge to the correct ground state solutions.

[0090] Unitary CC (UCC) may be more robust in MR cases since it is variational (i.e., does not suffer from non-variational breakdown) and involves substitutions within the occupied and virtual orbital spaces as well as substitutions from the virtual to the occupied orbital space. Unfortunately, UCC is incredibly costly to compute on classical computers and is therefore not even considered a viable alternative. On quantum devices, however, the unitary wave operator e'Tin UCC can be encoded rather efficiently using Trotterized products. Therefore, quantum devices employing a UCC QWFS as disclosed herein may provide an advantage over classical devices in scenarios where classical CC breaks down.

[0091] To investigate these potential benefits in terms of accuracy of the QWFS approach, the H4 benchmark system, representing a square lattice of four hydrogen atoms separated by 1.4 a.u. each, was investigated. The initial state of the QWFS circuits was chosen to reflect an appropriate multi-reference state. This was done by employing a simple circuit to entangle the involved bit strings and is shown in Figure 7. The collected results are portrayed in Table 3 below.

[0092] Table 3: Total energies of the H4 benchmark system in the STO-3G basis set for the classically exact Full CI and the CISD and CCSD(T) classical methods compared to QWFS (disclosed herein) results obtained from an ideal state vector simulator. The QWFS calculations were based on the second-quantized encoding scheme for the linear scaling k -UpCCGSD circuit ansatz employing k = 1,2,3.

[0093] Clearly, both of the classical approximate methods, CISD and CCSD(T), show deviations from the exact Full CI energies beyond the targeted chemical accuracy - noting that Full CI is computationally too expensive for practical applications. The QWFS approach disclosed herein, on the other hand, can reach chemical accuracy to Full CI starting with the 3-UpCCGSD circuit ansatz -therefore provides a viable alternative to the classical state-of-the-art.

[0094] It will be appreciated by persons skilled in the art that numerous variations and / or modifications may be made to the above-described embodiments, without departing from the broad general scope of the present disclosure. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Claims

CLAIMS:

1. A method for determining a characteristic of a wave function of electrons, the method comprising:(i) running a quantum circuit multiple times, the quantum circuit comprising multiple qubits and being configured to represent an eigenstate of a Hamiltonian of the electrons;(ii) after each time of running the quantum circuit, measuring a value of each of the multiple qubits to obtain multiple bit strings encoding respective basis functions;(iii) determining measurement statistics over the multiple bit strings;(iv) calculating, based on the measurement statistics and using a classical or an additional quantum computer, a wave function coefficient, including a sign value of each measurement, for each basis function encoded by the multiple bit strings;(v) calculating an energy value of the wave function based on the wave function coefficient for each basis function;(vi) repeating steps (i)-(v) and adjusting parameters of the quantum circuit to iteratively minimise the energy value to thereby improve the characteristic of the wave function.

2. The method of claim 1, wherein step (iv) of calculating the wave function coefficient comprises calculating, using the classical or the additional quantum computer, sign values for the measurements and adjusting the measurement statistics based on the sign values.

3. The method of claim 2, wherein calculating the sign values comprises calculating a sign value for each of the multiple bit strings.

4. The method of claim 3, wherein calculating the sign values is based on a classical or quantum machine learning model connected to the multiple bit strings of the quantum circuit obtained in step (ii).

5. The method of claim 2, 3 or 4, whereincalculating the sign value is based on multiple weights; and the method comprises before step (vi) repeating steps (iv)-(v) and adjusting the multiple weights using a classical or an additional quantum computer to iteratively minimise the energy value.

6. The method of any one of the preceding claims, wherein the characteristic comprises the wave function coefficient for each basis function or a binding energy provided by electrons.

7. The method of any of the preceding claims, wherein the method comprises calculating an initial value of the multiple qubits based on mean field electron interactions.

8. The method of any one of the preceding claims, wherein each bit string encodes one Slater determinant.

9. The method of claim 8, wherein the method further comprises filtering Slater determinants by keeping only Slater determinants with a minimum number of |1> values.

10. The method of any one of claims 1 to 7, wherein each bit string encodes a configuration state function represented by a linear combination of Slater determinants.

11. The method of any one of the preceding claims, wherein the each of the multiple bit strings is associated with an occupation number vector.

12. The method of any one of the preceding claims, wherein the method comprises: performing, in the quantum circuit parameterised single qubit gates and controlled multi-qubit gates; and adjusting parameters of the quantum circuit comprises adjusting the parameters of the single qubit gates.

13. The method of any one of the preceding claims, wherein running the quantum circuit comprises operating qubits integrated into a diamond substrate.

14. The method of any one of the preceding claims, wherein the method further comprises broadening the measurement statistics.

15. The method of claim 14, wherein broadening the measurement statistics comprises adding counts for bit strings that are in an equivalence class of a measured bit string.

16. A method implemented on a classical computer to determine a characteristic of a wave function of electrons, the method comprising:(i) for each of multiple runs of a quantum circuit, receiving a measured value of each of multiple qubits of the quantum circuit, to obtain multiple bit strings encoding respective basis functions;(ii) determining measurement statistics over the multiple bit strings;(iii) calculating, based on the measurement statistics, a wave function coefficient, including a sign value of each measurement, for each basis function encoded by the multiple bit strings; and(iv) calculating an energy value of the wave function based on the wave function coefficient for each basis function.

17. Software that, when executed on a classical computer, causes the classical computer to perform the method of claim 16.

18. A computer system comprising a classical processor configured to determine a characteristic of a wave function of electrons, by performing the method of claim 16.

19. A system, comprising: a quantum circuit, the quantum circuit comprising multiple qubits and being configured to represent an eigenstate of a Hamiltonian of the electrons; and a classical computer configured to perform the method of claim 16.

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