DE102020008010B4 - Procedure for reducing errors in a quantum computer - Google Patents

Abstract

It is already known that quantum computers can be used to simulate materials and molecules. However, quantum computers are prone to errors and have an intrinsic noise, which so far has made the real technical application of quantum computers impossible. Approaches are already known from the prior art which, despite the susceptibility to errors, make it possible to create meaningful simulations of quantum mechanical systems, but the errors still exist. The invention now allows, building on this, to reduce errors on the one hand and errors on the other to be included as part of the simulation. In addition, the invention makes it possible to inhibit the effect of intrinsic noise. This further improves the technical applicability of quantum computers for simulating materials and molecules.

Description

The present invention relates to a method for simulating a quantum mechanical system with the aid of a quantum computer which has a plurality of qubits, a quantum mechanical model of the quantum mechanical system first being mapped onto qubits of the quantum computer and being simulated thereon and, as part of an evaluation of the simulation, simulation results using measurements of the quantum computer can be extracted.

Such a method is already from the DE 10 2019 135 807 A1 previously known. The known solution provides that an image of the quantum mechanical system, for example a chemical structure, is selected in such a way that a disintegration of a qubit also leads to a physically meaningful system state. Although the simulation can deviate from a correct result, the system at least does not fall into an invalid, i.e. impossible, state.

On the basis of such considerations, however, work is continuing to make calculations with quantum computers more reliable. A quantum computer is a technically well-controlled quantum system, the calculation of which is based on the use of the laws of quantum mechanics. The basic unit of the quantum computer is the quantum bit, the so-called qubit. Like the well-known classic bit, the qubit can take the values 0 and 1. The main difference to the classic states is that the quantum memory can be in any superposition of the possible bit sequences. It follows that a quantum register of N qubits ^{encodes the information of 2 N} variables. A sufficiently large and well-functioning quantum computer can be used to solve certain mathematical problems that are unsolvable for classical computers. Such problems also include simulations of other quantum mechanical systems.

However, there are many technical difficulties in building a large quantum computer. The main difficulty is isolating a quantum computer from a noisy environment. Not all noise sources are known and as such are quantifiable. These also include non-equilibrium states of hardware materials, impurities that arise during the manufacturing process, local fluctuations that arise between different materials during the manufacturing process, and residual thermal excitations.

The effect of the noise on the quantum simulation can be characterized by its decoherence rate γ. This results in a characteristic time t _lost = 1 / γ in which the information stored in a qubit is lost. The noise and the corresponding decoherence rate are normally further divided into two specific parts, one part leading to the qubit decay at rate γ _dec, the other part leading to qubit dephasing at rate γ _dep . At present, the noise in superconducting quantum computers is dominated by qubit decay, while qubit dephasing takes place in ion trap quantum computers.

A very promising application of small quantum computers is the simulation of other quantum mechanical systems. In fact, it can be shown that quantum simulation algorithms can be faster than any classic computer, even for a small number of qubits.

From the publication of McArdle et al. (McArdle, Sam, Xiao Yuan, and Simon Benjamin. "Error-mitigated digital quantum simulation." Physical review letters 122.18 (2019) : 180501) a stabilizer-like method for the detection of depolarizing errors is already known. McArdle et al. shows that their method can significantly improve calculations that are subject to both stochastic and correlating noise, especially when combined with existing error mitigation methods.

Furthermore, the publication of Li et al. (Li, Ying, and Simon C. Benjamin. "Efficient variational quantum simulator incorporating active error minimization." Physical Review X 7.2 (2017) : 021050) a variation method that includes tightly integrated classical and quantum coprocessors. It is assumed that all operations in the quantum coprocessor are error-prone. The effects of such errors are minimized by artificially amplifying them and extrapolating them to the zero-error case. In comparison to a conventional, optimized trotting technique, this should make the protocol more efficient and generally more robust against the accumulation of errors.

Against this background, the present invention is based on the object of creating a method for simulating a quantum mechanical system which uses the advantages of quantum mechanics for the simulation, but at the same time reduces the susceptibility of these systems to errors.

This is achieved by a method for simulating a quantum mechanical system according to the features of independent claim 1. Sensible configurations of such a method can be found in the subsequent dependent claims.

According to the invention, it is provided that, in order to simulate a quantum mechanical system, this is first mapped onto the qubits of a quantum computer and then subsequently simulated on this. As part of an evaluation of this simulation, measurements are finally carried out on the qubits and the simulation results are thereby extracted from the quantum computer. In order to exclude errors as far as possible in this context, it is provided that known and new methods for error reduction complement one another, so that their effects lead to the most comprehensive possible reduction in the effect of various noise sources, including the intrinsic noise of the quantum computer itself.

In particular, three mechanisms are used for this, which are proposed and used in different variants. These are first of all error reduction through the clever choice of a qubit base, error reduction through extrapolation, in which the result of the simulation is measured at different error rates and the error-free result is determined through extrapolation, and temperature control through the introduction of an effective thermostat, whereby a or several qubits can be used to generate an effective temperature in the simulated system. Several of these procedures are combined with one another within the scope of the invention.

First of all, the solution according to the invention provides that when the quantum mechanical system is simulated, logical states of the qubits are mapped onto physical states of the qubits, this mapping being optimized to reduce errors.

Here, errors are minimized by optimizing logical qubit angles versus physical qubit angles in such a way that the inherent noise ideally drives the system to a solution, or at least to an approximate solution, of the problem. If an algorithm is used that repeatedly accesses the same gates, as is the case with the Trotter expansion in the quantum simulation, the self-noise, which can also be described as a type of friction, always becomes a stationary or quasi-stationary state in the system produce. Here, a situation is to be created in which the quasi-stationary state is as close as possible to the basic state of the simulated system. The logical qubits are selected accordingly.

Specifically, this can be solved by setting the logical states of the qubits using a transformation operator U (ϕ) = exp (-iϕσ _y / 2) = I · cos (ϕ2) - iσ _y sin (ϕ / 2) for any rotation angle ϕ can be mapped onto the physical states of the qubits, where I is the identity matrix and σ _{y is} a Pauli matrix in a space spanned by the states | 0〉 ≡ (1,0) and 11) ≡ (0,1).

$${|1\rangle }_{logical}={|1\rangle }_{physical}$$

Quantum computer providers allow access to their devices with a predefined definition for the state of each qubit. The two states are sometimes referred to as 0 and 1, or in a more physics-inspired notation as ↑ and ↓. The predefined setting for the qubits is known as the definition of the physical qubit. However, by superimposing the states of the physical qubit, two new states can be defined, which will be referred to below as the states of a logical qubit. The effect of the inherent noise of a quantum computer on logical qubits depends on the choice of logical qubits. The simplest choice is that the states of the simulated qubits, i.e. the logical qubits, are the states of the physical qubits $${|0〉}_{lOGical}={|0〉}_{pHysical}$$

$${|1〉}_{lOGical}={|1〉}_{pHysical}$$

$${|1\rangle }_{logical}={|0\rangle }_{physical}$$

Correspondingly, this can also be inverted to $${|0〉}_{lOGical}={|1〉}_{pHysical}$$

$${|1〉}_{lOGical}={|0〉}_{pHysical}$$

$${|1\rangle }_{logical}=\text{cos}\left(\varphi /2\right){|1\rangle }_{physical}-\text{sin}\left(\varphi /2\right){|0\rangle }_{physical}$$

More generally, any overlay can be chosen that is defined by a rotation angle ϕ, as shown below: $${|0〉}_{lOGical}=\text{cos}\left(\varphi /2\right){|0〉}_{pHysical}+\text{sin}\left(\varphi /2\right){|1〉}_{pHysical}$$

$${|1〉}_{lOGical}=\text{cos}\left(\varphi /2\right){|1〉}_{pHysical}-\text{sin}\left(\varphi /2\right){|0〉}_{pHysical}$$

This new base corresponds to a rotation that is defined by the transformation operator | 0/1〉 _logical = U (ϕ) | 0/1〉 _physical , where $$U\left(\varphi \right)=\text{exp}\left(-i\varphi {\sigma }_y/2\right)=\mathrm{I.}\cdot \text{cos}\left(\varphi /2\right)-i{\sigma }_y\text{sin}\left(\varphi /2\right).$$

Here I is the identity matrix and σ _{y is} a Pauli matrix in a space spanned by the states | 0〉 ≡ (1,0) and 11) ≡ (0,1). These types of single qbit operations are easy to implement on a quantum computer.

The ideal, noise-free time development in the simulation room always has the same form. A change in the logical qubit definitions is counteracted by 'opposing' changes in the physical gates used. However, the noise operators are fixed in physical space and therefore rotate in the simulation space. For example, the physical qubit decay changes to $${\sigma }^{+}\to U\left(\varphi \right){\sigma }^{+}{U}^{†}\left(\varphi \right)={\sigma }^{+}{\text{cos}}^2\left(\varphi /2\right)-{\sigma }^{-}{\text{sin}}^2\left(\varphi /2\right)-{\sigma }_z\text{cos}\left(\varphi /2\right)\text{sin}\left(\varphi /2\right)$$

In particular, when qubit states are tilted, i.e. ϕ = π is set, the physical qubit decay σ ^{+ is} mapped to the logical qubit excitation σ ^-. Furthermore, the dephasing operator transforms to $${\sigma }_z\to U\left(\varphi \right){\sigma }_z{U}^{†}\left(\varphi \right)={\sigma }_z\left({\text{cos}}^2\left(\varphi /2\right)-{\text{sin}}^2\left(\varphi /2\right)\right)+2{\sigma }_x\text{cos}\left(\varphi /2\right)\text{sin}\left(\varphi /2\right).$$

In particular, if qubit states are rotated by ϕ = π / 2, the qubit dephasing operator σ _z rotates to the depolarization _{operator σ x} .

In particular, the rotation angle ϕ can be selected for each qubit in such a way that the qubit decays into a state due to inherent noise of the quantum computer, which describes a meaningful physical system state of the simulated quantum mechanical system. The same applies to decay, with ideally in the context of mapping the quantum mechanical system to the qubits of the quantum computer the mapping is chosen so that each qubit has a meaningful physical state at least before or after the decay, but preferably both before and after the decay of the quantum mechanical system.

In parallel to the choice of the logical qubit angles, a suitable choice of the particle base can drastically reduce the effect of the noise. Here we consider a situation in which each physical or logical qubit describes an electronic orbital of the system. The effect of the self-noise on a quantum simulation depends on this choice.

In the representation of the second quantization of the electronic structure of a material, the electronic Hamilton function is projected onto orthogonal wave functions φ (x), which form the base states. These construct an electronic Hamilton function $$H={\displaystyle {\sum }_{pq}{H}_{pq}{c}_p^{†}{c}_q}+\frac{1}{2}{\displaystyle {\sum }_{pqrs}{H}_{pqrs}{c}_p^{†}{c}_q^{†}{c}_r{c}_s}.$$

der Elektronenerzeugungsoperator im Zustand i, und die Kinetik-Energie- und Nuklei-Wechselwirkungskoeffizienten sind durch Ein-Elektronen-Integrale $$h_{pq}={\displaystyle \int dx}{\phi }_p^{\ast }\left(x\right)\left(-\frac{{\nabla }^2}{2}-{\displaystyle {\sum }_I\frac{Z_I}{\left|r-R_I\right|}}\right){\phi }_q\left(x\right),$$

und die Elektron-Elektron-Coulomb-Wechselwirkungskoeffizienten durch Zwei-Elektronen-Integrale $$h_{pqrs}={\displaystyle \int d{x}_1}{\displaystyle \int d{x}_2}\frac{{\phi }_p^{\ast }\left(x_1\right){\phi }_q^{\ast }\left(x_2\right){\phi }_r\left(x_2\right){\phi }_s\left(x_1\right)}{\left|x_1-x_2\right|}$$

gegeben. Jede orthonormale Einzelteilchenbasis kann zur Beschreibung des Problems verwendet werden. Dies entspricht einheitlichen Transformationen von zweitquantisierten Systemoperatoren, c_i und ct. Eine optimale Wahl hilft uns, eine genauere Lösung (mit begrenzten Speicherressourcen) für untersuchte Systemeigenschaften, wie z.B. die Energie, zu finden. Diese Wahl, zusammen mit den logischen Qubit-Winkeln, kann auch dazu verwendet werden, Fehler drastisch zu mindern, wie unten beschrieben.Here, c _{i is} the electron annihilation and $c_i^{†}$

the electron generation operator in state i, and the kinetics energy and nuclei interaction coefficients are by one-electron integrals $$H_{pq}={\displaystyle \int dx}{\phi }_p^{\ast }\left(x\right)\left(-\frac{{\nabla }^2}{2}-{\displaystyle {\sum }_{\mathrm{I.}}\frac{Z_{\mathrm{I.}}}{\left|r-{\mathrm{R.}}_{\mathrm{I.}}\right|}}\right){\phi }_q\left(x\right),$$

and the electron-electron Coulomb interaction coefficients by two-electron integrals $$H_{pqrs}={\displaystyle \int d{x}_1}{\displaystyle \int d{x}_2}\frac{{\phi }_p^{\ast }\left(x_1\right){\phi }_q^{\ast }\left(x_2\right){\phi }_r\left(x_2\right){\phi }_s\left(x_1\right)}{\left|x_1-x_2\right|}$$

given. Any orthonormal single particle basis can be used to describe the problem. This corresponds to uniform transformations of second quantized system operators, c _i and ct. An optimal choice helps us to find a more precise solution (with limited storage resources) for the investigated system properties, such as energy. This choice, along with the logical qubit angles, can also be used to drastically reduce errors, as described below.

$$H_1={\displaystyle \sum _i{\omega }_i{c}_i^{†}{c}_i}$$

$$H_2={\displaystyle {\sum }_{pq}{\tilde{t}}_{pq}{c}_p^{†}{c}_q}+\frac{1}{2}{\displaystyle {\sum }_{pqrs}{h}_{pqrs}{c}_p^{†}{c}_q^{†}{c}_r{c}_s}.$$

An illustrative example is the approach based on the mean field Hamiltonian and its corrections. Here, the electronic Hamilton function is shown in the mean field base. This basis is first determined from classic simulations. $$H={\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}_2$$

$${\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}_1={\displaystyle \sum _i{\omega }_i{c}_i^{†}{c}_i}$$

$${\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}_2={\displaystyle {\sum }_{pq}{\tilde{t}}_{pq}{c}_p^{†}{c}_q}+\frac{1}{2}{\displaystyle {\sum }_{pqrs}{H}_{pqrs}{c}_p^{†}{c}_q^{†}{c}_r{c}_s}.$$

wobei |vac〉 das Vakuum, also der Zustand ohne Teilchen ist. Wir wissen also, dass es Orbital gibt die wahrscheinlich fast voll besetzt sind (solche mit ω_i< 0) und Orbitale, die fast leer sind (solche mit ω_i > 0 ). Im zweiten Bereich wird das System mit voller Wechselwirkung betrachtet und dadurch wird das System durch H₂ beschrieben. Elektronen können prinzipiell zwischen den beiden Bereichen hin und her springen und diesen Fall werden wir noch weiter unten diskutieren. Hier kann die Besetzung schwer vorhersehbar sein, aber im Mittel könne wir oft immer noch gute annahmen finden.Here the system description is divided into two areas. The middle field description is used for the range described by the Hamilton function H _1. As a result, this area is described solely by electronic orbitals. The Hamilton function H ₁ is diagonal because the orbitals were chosen that way. The ground state wave function is for this part $|G〉={\displaystyle {\prod }_{{\omega }_i<0}{c}_i^{†}}|vac〉,$

where | vac〉 is the vacuum, i.e. the state without particles. So we know that there are orbitals that are probably almost fully occupied (those with ω _i <0) and orbitals that are almost empty (those with ω _i > 0). In the second area, the system is considered with full interaction and the system is described by H ₂ . In principle, electrons can jump back and forth between the two areas and we will discuss this case further below. The occupation can be difficult to predict here, but on average we can often still find good assumptions.

It can be optimal to choose individual qubit states in such a way that they correspond to the individual particle states, i.e. that the same occupation probability is generated on average. In the presence of a physical qubit decay, it can be advantageous to choose logical angles in such a way that the decay drives the quantum simulation, preferably exclusively, in the direction of the midfield solution, i.e. that the decay produces the same occupancy probability that the orbital would have in the exact ground state. To achieve this, electronic particle states with positive energies, i.e. empty states, are represented by logical qubits with the rotation angles ϕ = 0 and electronic states with negative energies, i.e. filled states, by excited states with ϕ = π. In the worst case of a strong decay-defined solution, there is again a midfield solution. For a state with an occupation of ½, which can occur in the area with interaction, one would choose the angle ϕ = π / 2.

The occupation probabilities in the interacting area, which is described by H ₂ , can also look more complicated. This can create all kinds of partial occupations. In order to produce this in a qubit that suffers from decay, the angle of rotation must be selected in the range 0 <ϕ <π.

However, the particle base can also be chosen in some other way, in particular by a choice that helps to find a more accurate solution for part of the system or for a different observable region than the energy. Suitable choices could be localized states in the vicinity of certain regions of the system or eigenstates of other objects of observation. Such conditions are usually characterized by a strong partial filling, even in the mean field solution.

A problem is considered with a few strongly active, partially filled states, a set of mostly filled states that are energetically below these states, and a lot of mostly empty states that are energetically above these states. In the presence of physical decay it is then probably optimal to choose mostly empty electronic states, which are represented by the angle ϕ = 0 and mostly filled electronic states by excited states by ϕ = π.

The states in the active region are either partially filled or empty. Partial filling can be achieved by choosing the angles 0 <ϕ <π.

For example, a physical qubit decay to the ground state with the logical qubit angle ϕ corresponds to the electronic occupation $$〈n〉={\text{sin}}^2\left(\varphi /2\right).$$

Partial filling driven by decay can be advantageous in a cluster-bath approach, for example, if there is interest in the properties of localized states in the vicinity of the cluster. For the self-consistent cluster-bath model of solids, too, one of the main problems is the adaptation of the kinetic energies at the boundary between cluster and bath.

Such a cluster-bath approach can be of particular advantage for the method according to the invention, which is why it will be discussed in more detail. In the cluster-bad approach, the quantum mechanical system to be simulated is divided into a cluster and a bath. Some orbitals are defined as part of the bath and some as part of the cluster. When mapping the orbitals onto qubits, the qubits are first evaluated with regard to their system properties and categorized into high-performance qubits and low-performance qubits. The high-performance qubits are then assigned to the orbitals in the cluster and the low-performance qubits are assigned to the orbitals in the bathroom.

$$H_C={\displaystyle \sum _{pq\in cluster}{t}_{pq}{c}_p^{†}{c}_q}+\frac{1}{2}{\displaystyle \sum _{pqrs\in cluster}{h}_{pqrs}{c}_p^{†}{c}_q^{†}{c}_r{c}_s}$$

$$H_B={\displaystyle \sum _{i\in bath}{\omega }_i{c}_i^{†}{c}_i}$$

$$H_I={\displaystyle \sum _{p\in cluster,i\in bath}{t}_{pi}{c}_p^{†}{c}_i+t_{ip}{c}_i^{†}{c}_p}$$

In this model we assume a Hamilton function as follows. $$H=H_{\mathrm{C.}}+H_{\mathrm{B.}}+{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}_1$$

$$H_{\mathrm{C.}}={\displaystyle \sum _{pq\in cluster}{t}_{pq}{c}_p^{†}{c}_q}+\frac{1}{2}{\displaystyle \sum _{pqrs\in cluster}{H}_{pqrs}{c}_p^{†}{c}_q^{†}{c}_r{c}_s}$$

$$H_{\mathrm{B.}}={\displaystyle \sum _{i\in batH}{\omega }_i{c}_i^{†}{c}_i}$$

$$H_{\mathrm{I.}}={\displaystyle \sum _{p\in cluster,i\in batH}{t}_{pi}{c}_p^{†}{c}_i+t_{ip}{c}_i^{†}{c}_p}$$

The system description is divided into three parts: Cluster H _C , which describes electrons completely with interaction, a bath H _{B of} non-interacting electrons and electron hopping between the two areas H _I. The modeling within the cluster is as precise as possible, while the bathroom and the Interaction in the midfield approach must be taken into account. This provides the most detailed solvable description of many solids and is the basis of the dynamic mean field theory (DMFT).

$$S_{p\in cluster}^{+}\left(\omega \right)={\displaystyle \sum _{i\in filledbath}{t}_i^2\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega +{\omega }_i\right)}^2}}$$

In the presence of intrinsic bad qubit decay, this noise can be used to drive the system towards complete solution, i.e. towards the ground state. A system of interaction-free electrons is fully characterized by a spectral function. Accordingly, the qubits that model the bath must have the same spectral function as the electronic system to be simulated. The qubits in the cluster see the following spectral function when they are effectively coupled to the bad qubits: $${\mathrm{S.}}_{p\in cluster}^{-}\left(\omega \right)={\displaystyle \sum _{i\in emptybat\mathrm{<figure-callout\; id="2"\; label="H\; t\; i"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}{t}_i^{}{}_2^{}\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega -{\omega }_i\right)}^2}}$$

$${\mathrm{S.}}_{p\in cluster}^{+}\left(\omega \right)={\displaystyle \sum _{i\in filledbat\mathrm{<figure-callout\; id="2"\; label="H\; t\; i"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}{t}_i^{}{}_2^{}\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega +{\omega }_i\right)}^2}}$$

Here γ _{j is} the decay rate of the bad qubit j. One part of the spectral density describes empty states (-) and the other part describes filled states (+) in the bathroom when they do not interact with the cluster. The angles of the logical qubits in relation to the physical qubits are chosen in the same way as above, empty spaces with ϕ = 0 and filled spaces with ϕ = π. The bad decay then drives the system to its full solution. The spectral function can be optimized by choosing the coupling parameters t _i so that it corresponds to the spectral function of the system to be simulated.

It was implicitly assumed above that the self-noise of the cluster qubits is negligible. If this is not the case, the noise from the cluster qubits is driving the system to a wrong solution. However, this mistake can be mitigated. For this purpose it can be advantageous to use cluster angles 0 <ϕ <π in order to obtain only partially occupied cluster states in the case of noise.

In an advantageous development, the method can be supplemented by extrapolating the simulation results into a low-noise, preferably noise-free, environment during the simulation of the quantum mechanical system.

This method reduces the effect of noise by extrapolating the noisy result into a low-noise environment. This idea is based on the ability to effectively increase the rate of decoherence. It is assumed that the expected value of the variable X that the noisy quantum computer receives can be written as a Taylor expansion. $$〈X〉\left(\gamma \right)=〈X〉\left(0\right)+{\displaystyle \sum _{k=1}^n{a}_k{\gamma }^k}$$

Here 〈X〉 (0) relates to the noise-free result, i.e. γ = 0. The result of a calculation on a quantum computer gives 〈X〉 (γ ₀ ), where γ _{0 is} the normal decoherence rate.

The simplest estimate for the error-free result is obtained using a linear extrapolation. The quantum simulation is carried out in the presence of two noise rates, γ ₁ and γ ₀ , and only the expansion of the leading order, n = 1, is considered. The front factor in the linear term has the form $$a_1=\frac{〈X〉\left({\gamma }_1\right)-〈X〉\left({\gamma }_0\right)}{{\gamma }_1-{\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="DE102020008010B4\_0043.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_0}.$$

So is $$〈X〉\left(0\right)\approx 〈X〉\left({\gamma }_0\right)-a_1{\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="DE102020008010B4\_0043.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_0.$$

durchgeführt, wobei i ∈ [0,1,2, ...,n]. Eine neue Schätzung/Extrapolation wird erhalten durch $$\langle X\rangle \left(0\right)\approx {\displaystyle \sum _{i=0}^n{\mu }_i\langle X\rangle \left({\epsilon }_i\right)}$$

This linear estimate can be further improved by including higher order corrections. For this purpose, the quantum simulation is used with several different decoherence rates $${\epsilon }_i=c_i\gamma$$

performed, where i ∈ [0,1,2, ..., n]. A new estimate / extrapolation is obtained by $$〈X〉\left(0\right)\approx {\displaystyle \sum _{i=0}^n{\mu }_i〈X〉\left({\epsilon }_i\right)}$$

The coefficients must meet the following conditions for k ∈
[0, 1, 2, ..., n]. $${\displaystyle \sum _{i=0}^n{\mu }_i}=\mathrm{1,}{\displaystyle \sum _{i=0}^n{\mu }_i{c}_i^k}=\mathrm{0,}\to {\mu }_i={\displaystyle \prod _{m\ne i}\frac{c_m}{c_m-c_i}}$$

Such a choice of the coefficients cancels the n-th order error. The idea of extrapolation can also be extended to the exponential approach.

In order to extrapolate a noise-free state, we need to be able to generate artificial noise of different strengths in our quantum computer. A well-known method to achieve this is to simply perform all arithmetic operations on the quantum computer more slowly. This effectively increases the decoherence during a calculation. However, another method will be discussed here.

Changing the speed of operations is difficult to perform and requires detailed control over the quantum computer. One method that is easy to control is the effect of increasing the physical time of a trotter step dτ.

The effect of the physical trotting time dτ on the error rate γ can be derived from the following derivation. The ideal quantum computer time evolution is defined by the Hamilton function H _QC (t). The coupling to the external noise is described by the Hamilton function H _E. Assuming that dτ << 1 / γ, the time evolution operator over the simulation time dt, taking into account the physical time dt, can be written as $$\begin{array}{ll}{U}_{tOtal}\left(dt,d\tau \right)\hfill & \approx U_{Q\mathrm{C.}}\left(dt\right)U_{\mathrm{E.}}\left(d\tau \right)=U_{Q\mathrm{C.}}\left(dt\right)exp\left[id\tau H_{\mathrm{E.}}\right]\hfill \\ \hfill & =U_{Q\mathrm{C.}}\left(dt\right)exp\left[idt\times c{H}_{\mathrm{E.}}\right]\hfill \end{array}$$

Here c = dτ / dt, which results in a relationship between the time steps in the hardware and in the simulated system. After the factor c is multiplied by the neighborhood Hamilton function, it can simply be included in the definition of H _E , with $$H_{\mathrm{E.}}\to c{H}_{\mathrm{E.}}$$

This is what the neighborhood Hamilton function looks like in logical space. Now all energies and coupling terms that relate to the environment are multiplied by c. It follows that the decoherence rate of the logical qubits changes in the same way in the quantum simulation.

As already mentioned above, the increase in the physical trotting time dτ can itself be achieved by simple longer gate control pulses. However, this facility may not be possible or available to the user. In this case, the trotting time can be increased by several equivalent operations within one trotting step. A simple example is the division of the gate operation exp (iφσ _z ) into exp (iφσ _z / 2) exp (iφσ _z / 2). This would double the physical time required to complete the entire operation. These changes effectively lead to a doubling of the decoherence rate for the simulation, which means that the extrapolation to a noise-free state can then be carried out.

In a further development of the method, an increase in the decoherence rate can be generated artificially by means of additional qubits. In this case, a trotter step is first carried out iteratively and then an excitation exchange is carried out between a bad qubit and an auxiliary qubit with a probability p << 1. The auxiliary qubit is measured and in the event that the auxiliary qubit was measured in its excited state, it is reset to its basic state. Then finally the trotter step is started again. This method allows the decoherence times of the qubits to be set and, finally, the simulation results can also be transformed into a low-noise, preferably noise-free, state by extrapolating the decoherence time and deriving a transformation rule, as well as applying it to the simulation results of the qubits.

The possibility of artificially creating decoherence will be explained in detail here. A gate-based quantum development is constructed by successive applications of identical trotter steps, each of which represents a small time development over the simulation time dt. Such a digitized time evolution can also be modified in order to effectively reproduce the decay or dephasing of selected qubits. This enables the noise extrapolation to be performed much more rigorously.

1. (i) Anwendungszeit-Evolutions-Operation U_QC(dt), also der ursprüngliche Trotter-Schritt,
2. (ii) Anwendung eines Anregungsaustauschs zwischen einem Bad-Qubit und einem diesem zugeordneten Hilfs-Qubit mit einer Wahrscheinlichkeit p << 1,
3. (iii) Messung des Hilfs-Qubits,
4. (iv) wenn sich das Hilfs-Qubit im angeregten Zustand befand, wird es in seinen Grundzustand zurückversetzt. Dann wird ab Schritt (i) iteriert.

An example of the creation of artificial decay in the trotterized time evolution is described below. Each trotter step is divided into four steps:

1. (i) Application time evolution operation U _QC (dt), i.e. the original trotter step,
2. (ii) Application of an excitation exchange between a bad qubit and an auxiliary qubit assigned to it with a probability p << 1,
3. (iii) Measurement of the auxiliary qubit,
4. (iv) if the auxiliary qubit was in the excited state, it is reset to its ground state. It is then iterated from step (i).

wobei der Lindblad-Superoperator, bzw. Dissipator, wie folgt eingeführt werden kann. $$\mathcal{L}\left[\widehat{D}\right]\equiv p\left({\sigma }^{+}\widehat{D}{\sigma }^{-}-\frac{1}{2}{\sigma }^{-}{\sigma }^{+}\widehat{D}-\frac{1}{2}\widehat{D}{\sigma }^{-}{\sigma }^{+}\right)$$

It can be shown that the density matrix D̂ of the simulated system changes with each measurement process $$\widehat{\mathrm{D.}}\to \widehat{\mathrm{D.}}+\mathcal{L.}\left[\widehat{\mathrm{D.}}\right],$$

where the Lindblad super operator, or dissipator, can be introduced as follows. $$\mathcal{L.}\left[\widehat{\mathrm{D.}}\right]\equiv p\left({\sigma }^{+}\widehat{\mathrm{D.}}{\sigma }^{-}-\frac{1}{2}{\sigma }^{-}{\sigma }^{+}\widehat{\mathrm{D.}}-\frac{1}{2}\widehat{\mathrm{D.}}{\sigma }^{-}{\sigma }^{+}\right)$$

Here σ ^{+ (-) is} a spin-drop / increase operator, which corresponds to the excitation-generation / annihilation operator. Repeatedly performing similar entanglement operations and measurements then creates an artificial decay at the rate γ = p / dt. This rate can then be controlled by the commutation probability p. This in turn allows a noise extrapolation that is only related to the decay, as described above.

In addition to decay, other effects can be achieved by any rotation of the original dissipator operator. In particular, the choice ϕ = π simulates the noise that leads to the excitation of the qubit. Similarly, a finite temperature bath can also be realized if a statistical distribution is used for the rotation of the dissipator operator.

Intrinsic dephasing can also be created artificially. One possibility for this is to use the above procedure, but with the dissipator operator σ _z . Another approach is to establish that the dephasing corresponds to the fluctuations of the qubit frequencies and that the fluctuations of a qubit frequency in turn correspond to the terms of the type δωσ _z in the Hamilton function. In order to increase the size of the dephasing alone, one can therefore also insert artificial fluctuations of the corresponding Hamilton terms in each Trotter step, with corresponding statistics. In particular, this method would enable any time correlation of the noise to be reproduced.

The last possibility of error correction discussed here provides for temperature regulation to be carried out to produce a specific heat distribution within the quantum computer by putting at least one auxiliary qubit into an excited state for the purpose of absorbing thermal energy.

This is necessary because the qubit dephasing can cause the system to heat up. If the heating is undesirable, it can be mitigated by the introduction of a thermostat between the simulation and the physical space, as will be explained below.

As already mentioned above, the time evolution of a quantum system is to be simulated under the effect of the Hamilton function H. Then the effects of self-noise can be used to bring the system into a quasi-stationary state. In order to achieve a specific heat distribution, an additional thermostat can be added to the Hamilton function, for example $$H\to H+H_{tHermO}$$

sein, wobei ${\sigma }_z^{sim}$

auf den Simulationsraum wirkt und ${\sigma }_x^{aux}$

auf das Hilfs-Qubit.A specific example for H _thermo is a Hamilton function, which acts on the simulation space in such a way that the number of excitations is retained. Maintaining the number of excitations is necessary if an electronic structure problem is to be simulated because the number of excitations corresponds to the number of electrons and the number of electrons is fixed. The Hamilton function connects the simulation space with an operation outside the simulation space, i.e. with an auxiliary qubit. The operation on the auxiliary qubit must be able to put the qubit from the ground state into an excited state. If the auxiliary qubit has an intrinsic decay rate / decay rate and is at zero temperature, then the auxiliary qubit will absorb energy at and around the energy gap ΔE between the ground state and the excited state. A Hamilton function of this kind could be $$H_{tHermO}\propto {\sigma }_z^{sim}{\sigma }_x^{aux}$$

be, where ${\sigma }_z^{sim}$

acts on the simulation space and ${\sigma }_x^{aux}$

on the auxiliary qubit.

The invention described above is explained in more detail below using an exemplary embodiment.

• 1 eine schematische Darstellung der Quantensimulation mit allen erfindungsgemäßen Fehlerkorrekturverfahren,
• 2 eine schematische Darstellung einer optimalen Abbildung logischer Qubits auf physische Qubits mit einem geeigneten Rotationswinkel, sowie
• 3 eine Darstellung einer linearen Rauschextrapolation in einem Schaubild mit (X) über γ sowie einem Hilfsschaubild mit γ über dτ.

Show it

• 1 a schematic representation of the quantum simulation with all error correction methods according to the invention,
• 2 a schematic representation of an optimal mapping of logical qubits to physical qubits with a suitable angle of rotation, and
• 3 a representation of a linear noise extrapolation in a graph with (X) over γ and an auxiliary graph with γ over dτ.

1 shows a schematic representation of a simulation 1 a quantum mechanical system, such as a chemical structure, with three different error correction methods, which are used in addition to one another.

The central area in the middle visualizes the simulation 1 in the overall process within a logical space 13th . The method works best for quantum simulation algorithms that aim for a steady state of the system. A noise source in a physical space feeds noise from the outside 2 into the simulation 1 a. An interface between physical qubits and logical qubits of the simulation space introduces a rotation angle ϕ between the two spaces and affects the noise seen by the quantum simulation. An optimal angle ϕ minimizes the effect of the noise. In the simulation, the noise-induced heating can be controlled by adding Hamilton functions, which regulate the temperature 5 introduce, be reduced. Before the final low-noise simulation results 10 can be obtained is an extrapolation 4th carried out, in which the simulation is carried out several times, each with different external noise and, if necessary, with additional artificial noise 11 to resolve contributions from various sources of error more precisely.

$$S_{p\in cluster}^{+}\left(\omega \right)={\displaystyle \sum _{i\in filledbath}{t}_i^2\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega +{\omega }_i\right)}^2}}$$

entspricht. Bei Vorhandensein von Cluster-Qubit-Zerfall sind die Cluster-Qubit-Winkel die zu einer Teilfüllungen führen vorteilhaft. 2 shows an example of optimal angles in a cluster-bad quantum simulation in the presence of intrinsic decay. The qubit decay drives the physical qubits towards their ground states. By introducing an angle ϕ = π between the right-hand physical and logical bad qubits 7th the corresponding electronic states are driven towards complete filling. In the absence of cluster qubit decay, this mechanism drives the cluster bad simulation to a solution that reduces the bad spectral densities $${\mathrm{S.}}_{p\in cluster}^{-}\left(\omega \right)={\displaystyle \sum _{i\in emptybat\mathrm{<figure-callout\; id="2"\; label="H\; t\; i"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}{t}_i^{}{}_2^{}\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega -{\omega }_i\right)}^2}}$$

$${\mathrm{S.}}_{p\in cluster}^{+}\left(\omega \right)={\displaystyle \sum _{i\in filledbat\mathrm{<figure-callout\; id="2"\; label="H\; t\; i"\; filenames="DE102020008010B4\_0041.png"\; state="\{\{state\}\}">H</figure-callout>}}{t}_i^{}{}_2^{}\frac{{\gamma }_i/2}{{\left({\gamma }_i/2\right)}^2+{\left(\omega +{\omega }_i\right)}^2}}$$

is equivalent to. In the presence of cluster qubit decay, the cluster qubit angles that lead to partial filling are advantageous.

3 finally shows a linear noise extrapolation of the expected value (X). The noise-free result (X) (0) can be estimated by measuring the expected values under noise, (X) (y> 0). The linear estimate includes measurements under two noise quantities, γ ₀ and γ ₁ . The noise 2 can be increased by increasing the Trotter step time dτ or by increasing the probability of artificial decay p. The linear estimate can be improved with more measurements and non-linear extrapolation.

A method for simulating a quantum mechanical system is thus described above, which uses the advantages of quantum mechanics for the simulation, but at the same time reduces the susceptibility of these systems to errors.

List of reference symbols

1

simulation

2

rush

3

Illustration

4th

Extrapolation

5

Temperature control

6th

Cluster qubits

7th

Bad qubits

8th

Auxiliary qubits

9

Surroundings

10

Simulation results

11

artificial noise

12th

Physical space

13th

Logical space

Claims (9)

Method for simulating a quantum mechanical system with the aid of a quantum computer, which has a plurality of qubits (6, 7, 8), whereby a quantum mechanical model of the quantum mechanical system is first mapped onto qubits (6, 7, 8) of the quantum computer and is simulated on this, and As part of an evaluation of the simulation (1), simulation results (10) are extracted by measurements of the quantum computer, the simulation results (10) being influenced by several methods used in addition to one another for error reduction, which reduce the effects of noise (2) acting on the quantum computer , characterized in that in the simulation (1) of the In the quantum mechanical system, a mapping (3) of logical states of the qubits (6, 7, 8) to physical states of the qubits takes place, whereby this mapping (3) is optimized to reduce errors, and that the quantum mechanical system is divided into a cluster and a bath , whereby the qubits (6, 7, 8) are initially evaluated with regard to their system properties and categorized into high-performance qubits (6) and low-performance qubits (7, 8) and, as part of a mapping process, the high-performance qubits (6) are assigned to simulate the cluster and the low-performance qubits (7) are assigned to simulate the bath. Procedure according to Claim 1 , characterized in that the logical states of the qubits each with the help of a transformation operator U (ϕ) = exp (-iϕσ _y / 2) = I · cos (ϕ / 2) - iσ _y sin (ϕ / 2) for any rotation angle ϕ the physical states of the qubits (6, 7, 8) are mapped, where I is the identity matrix and σ _{y is} a Pauli matrix in one of the states | 0〉 ≡ (1,0) and | 1〉 ≡ (0,1) spanned space is. Procedure according to Claim 2 , characterized in that the angle of rotation ϕ for each qubit (6, 7, 8) is chosen so that the qubit (6, 7, 8) decays due to the inherent noise of the quantum computer into a state which is a reasonable physical system state of the simulated quantum mechanical Systems describes. Method according to one of the Claims 1 until 3 , characterized in that in the context of the mapping (3) of the quantum mechanical system to the qubits (6, 7, 8) of the quantum computer, the mapping (3) is chosen so that each qubit (6, 7, 8) has a meaningful physical single-particle state of the quantum mechanical system. Procedure according to Claim 4 , characterized in that the mapping (3) of the quantum mechanical system to the qubits (6, 7, 8) of the quantum computer is chosen so that each qubit (6, 7, 8) continues to have a meaningful physical single-particle state of the quantum mechanical system after a decay describes. Method according to one of the preceding claims, characterized in that during the simulation (1) of the quantum mechanical system, the simulation results (10) are extrapolated into a low-noise, preferably noise-free, environment. Procedure according to Claim 6 , characterized in that an artificial noise (11) of different strength is iteratively generated in the quantum computer for extrapolation of a noise-free state and in each case decoherence times of the qubits (6, 7, 8) are measured as a function of the artificial noise (11), with a linear relationship between the artificial noise (11) and the decoherence times is determined and from this a transformation rule for a transformation of simulation results (10) into a low-noise, preferably noise-free, environment is derived and finally the simulation results (10) are transformed with the derived transformation rule. Procedure according to Claim 7 , characterized in that an extrapolation (4) is carried out by iteratively first performing a trotter step, then performing an excitation exchange between a bad qubit (7) and an auxiliary qubit (8) with a probability p << 1 , the auxiliary qubit (8) is measured and, in the event that the auxiliary qubit (8) was measured in its excited state, it is reset to its basic state and finally the trotter step is started again, based on the measurements the decoherence times of the qubit (7, 8) is changed and finally the simulation results (10) by extrapolation (4) of the decoherence time and derivation of a transformation rule, as well as its application to the simulation results of the qubit (7, 8) in a low-noise, preferably noise-free, State to be transformed. Method according to one of the preceding claims, characterized in that a temperature control (5) for producing a specific distribution function is carried out within the quantum computer by putting at least one auxiliary qubit (8) into an excited state to absorb thermal energy.

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