EP4205044A1

EP4205044A1 - Method for operating a circuit having a first and a second qubit

Info

Publication number: EP4205044A1
Application number: EP21763382.5A
Authority: EP
Inventors: Mohammed Ansari; Xuexin Xu
Original assignee: Forschungszentrum Juelich GmbH
Current assignee: Forschungszentrum Juelich GmbH
Priority date: 2020-08-26
Filing date: 2021-08-24
Publication date: 2023-07-05
Also published as: US20230318600A1; CN115867924A; WO2022043297A1; JP2023540060A

Abstract

The invention relates to a method for operating a circuit having a first qubit (7) and a second qubit (3), which circuit is designed so that the frequency of the first qubit (7) differs from the frequency of the second qubit (3), and having a coupler (4) which couples the first qubit (7) and the second qubit (3), a cross-resonance pulse being transmitted to the first qubit (7), and the amplitude of the cross-resonance pulse being selected so that the two-qubit phase error is minimal or at least substantially minimal in terms of amount. The two-qubit phase error is determined by measuring the qubit Hamiltonian value and measuring the coupling strength of the ZZ interaction in kilohertz precision. A high two-qubit gate fidelity can be achieved by the invention.

Description

Method of operating a circuit with a first and a second qubit

The invention relates to a method for operating a circuit with a first and a second qubit and with a coupler which couples the first qubit to the second qubit.

A classic computer can store and process data in the form of bits. Instead of bits, a quantum computer stores and processes quantum bits, also called qubits.

Like a bit, a qubit can assume two different states. The two different states can be two different energy eigenvalues, which can represent a 0 and a 1 as in the classical computer. The ground state, i.e. the lowest energy level, can be represented by 0. The notation IO> can be used for this. The state with the next higher energy can be provided for the 1, which can be expressed by the notation I1>. In addition to these two basic states IO> and I1>, a qubit can assume the states IO> and I1> at the same time. Such a superimposition of the two states IO> and 11> is called superposition. This can be described mathematically by l\|/ ^{> =} Co IO> + Ci 11 >. A superposition can only be maintained for a very short time. There is therefore only very little time available for arithmetic operations using superposition. Physical qubits made in the lab will not only have these two states 0 and 1, also called computational states, they will also have higher levels of excitation, labeled I2>, I3>, I4>... will. The higher excitation levels are also referred to as non-calculation states.

Qubits of a quantum computer can be independent of each other. However, qubits can also be dependent on one another. The dependent state is called entanglement.

In a quantum computer, several qubits are combined into a quantum register. For a register of two qubits there are then the base states 100> , 101>, I10>, 111>. The state of the register can be any superposition of the base states of a register. Two qubits define a computational state 100>, 101>, 110>, 111>. The number of calculation states for n qubits is 2 to the power of n, ie 2n. Two qubits define non-calculation states, such as IO2>, IO3>, ...,I2O>, I3O>,..., I12>, I13>, I22>, I31>, .... The number of non- Calculation states can be large and even infinite.

A circuit with two qubits includes energy levels. If the two qubits are in the state |n1 ,n2>, the corresponding energy level is En1n2. n1 and n2 are states of the first qubit and the second qubit, respectively. E11 is therefore the energy level of the circuit when the state |1 ,1 > is present, i.e. both qubits are in the state |1 >. For a qubit, the energy difference between states 0 and 1 is denoted as the qubit frequency ( =(E1-E0)/h, where h is Planck's constant. The energy spectrum of a qubit is not equidistant (evenly distributed). Hence, the energy spectrum is of a qubit not similar to that of a harmonic oscillator The qubit anharmonicity is defined as <5=(E2-2E1-E0)/h.

In the case of non-interacting qubits 1 and 2, the energy levels of the computational states are the lowest four levels, and all non-computational states have energies greater than E11. In the case of the interacting qubits 1 and 2, some of the non-computational states can find energies below E11. This depends on the strength of the interaction between the qubits and also on the qubit frequencies and anharmonicities.

In a quantum computer, there are both entangled qubits and qubits that are independent of each other. Ideally, independent qubits do not affect each other. The independent qubits are called idle qubits. In the absence of a gate, superconducting open-loop qubits accumulate errors in the phase of two qubit states. When the state of two qubits is the same, both 0 or both 1, they accumulate a positive phase. When the states of two qubits differ from each other, they accumulate a negative phase. This means that in the absence of a gate, the idle state |00> changes to exp(+i g.t) |00> after time t. Similarly, the |11> state changes to exp(+i g.t)|11 >. The state |01> evolves into exp(-i g.t) |01 >. The state |10> evolves to exp(-i g.t) |10>.

Two superconducting qubit gates are always accompanied by a degrading effect of an unwanted ZZ-type interaction. In the absence of the gate this ZZ interaction appears with a coupling strength proportional to g. This is the same coefficient that causes a two-qubit state phase error.

If an action is applied to a quantum register over time, this is called a quantum gate or gate. A quantum gate therefore acts on a quantum register and thereby changes the state of a quantum register. A quantum gate that is essential for quantum computers is the CNOT gate. If a quantum register consists of two qubits, the first qubit acts as the control qubit and the second as the target qubit. The CNOT gate causes the ground state of the target qubit to change when the ground state of the control qubit is I1>. The ground state of the target qubit does not change if the ground state of the control qubit is IO>.

The CNOT gate is an example of a two-qubit gate applied to entangle two interacting qubits. Applying CNOT to a two-qubit state where the first qubit is |0> leads to the same state. Applying CNOT to the first qubit in state |1> results in a state reversal, changing |0> to |1> or |1> to |0> in the second qubit.

Applying CNOT to qubits that have higher excitation levels above states 0 and 1 does not only result in a two-qubit phase error for the final state. This suggests that during the time CNOT was applied, the state accumulated a phase due to an unwanted ZZ interaction between qubits.

The presence of two-qubit phase errors between the first and second qubit and their crosstalk is one of the main problems of quantum computers. In superconducting qubits, such unwanted entanglements exist due to the presence of higher excitation energies in each qubit. Superconducting qubits, such as transmons, undesirably exchange information and energy via non-computational states and energy levels. Such an interaction between computational states and non-computational states is a ZZ interaction. The ZZ interaction is always present whether any gate is applied to qubits or not. The ZZ interaction in the absence of a gate is called static ZZ interaction. The coupling strength of the static ZZ interaction g is equivalent to the following energy difference: E11-E01-E10+E00. This amount corresponds to Level repulsion in computational states from their interaction with non-computational states. Level repulsion is also known as avoided superposition. Static repulsion is always present, causing qubits to accumulate erratic phases while at rest.

When a microwave is applied to one of two qubits, all energy levels En1 n2 of the two qubits change, some decrease and some increase. This leads to the desired two-qubit gate, such as CNOT.

Applying microwave two-qubit gates changes the repulsion levels of non-computational energy levels. The gate changes the magnitude of the phase error from exp(±i g.t) in free qubits to y exp(±i y.t) in the presence of a microwave pulse. The magnitude of the phase error can increase or decrease. Eliminating the level repulsion sets y=0 and this removes the phase error exp(±i Y-t)> by setting it to 1. This process of producing a "phase-error-free two-qubit state" is the subject of this invention.

Document WQ2014/140943A1 discloses a device with at least two qubits. A bus resonator is coupled to the two qubits. A transmon and a CSFQ (capacitively shunted flux qubit) are given as examples of qubits. The publications WO 2013/126120 A1 and WO 2018/177577 A1 disclose a transmon or a CSFQ as examples of qubits. The publication "Engineering Cross Resonance Interaction in Multi-modal Quantum Circuits, Sumeru Hazra et. al.,, arXiv:1912.10953v1 [quant-ph] 23 Dec 2019”, discloses tuning of cross-resonance interactions for a multi-qubit gate. Cross-resonance pulses are known from this publication. The publication US 2014264285 A discloses a quantum computer with at least two qubits and a resonator. The resonator is coupled to the two qubits. There is a microwave drive. A tuned microwave signal applied to a qubit can activate a 2-qubit phase interaction. The publication US 2018/0225586 A1 discloses a system with a superconducting control qubit and a superconducting target qubit.

The publication "Suppression of Unwanted ZZ Interactions in a Hybrid Two-Qubit System, Jaseung Ku, Xuexin Xu, Markus Brink, David C. McKay, Jared B. Hertzberg, Mohammad H. Ansari, and BLT Plourde, arXiv:2003.02775v2 [quant -ph] 9 Apr 2020” reveals a Suppression of unwanted ZZ interactions using a circuit that includes two qubits. The first qubit is a qubit with a negatively anharmonic energy spectrum. The second qubit is a qubit with a positively anharmonic energy spectrum. This reference shows circuit features for setting the open circuit two qubit phase error to zero, ie g=0.

It is an object of the invention to improve two-qubit gate fidelity. The two-qubit gate fidelity determines to what extent the final state of two qubits after application of a real gate is similar to the final state after application of an ideal gate. In this invention we eliminate the two qubit phase error from a two qubit gate and improve the gate fidelity.

The object of the invention is achieved by a method having the features of the first claim. Advantageous configurations result from the dependent claims.

To solve the task, a circuit includes a first qubit and a second qubit. The frequency of the first qubit is different from the frequency of the second qubit. The anharmonicity of the two qubits can have the same or opposite sign. There is a coupler that couples the first qubit and the second qubit. There is at least one microwave generator that can be used to generate microwaves. The microwave generator is coupled to the first qubit such that microwave pulses can be sent to the first qubit. A first cross-resonance pulse is sent to the first qubit. The amplitude of the first cross-resonance pulse is adjusted such that the magnitude of the two-qubit phase error that occurs after application of a cross-resonance pulse for duration t becomes significantly smaller. Preferably, the CR-induced two-qubit state phase error becomes exactly zero for the duration t that the cross-resonance pulse is applied.

How the height of the amplitude of the cross-resonance pulse is to be selected can be determined theoretically, for example, by means of the QED circuit theory. In order to be able to determine experimentally whether the repulsion of the CR-induced level from non-computing states is zero or at least close to zero, a modified version of the method of quantum Hamilton tomography can be used. The standard method of quantum Hamiltonian tomography can be found in the publication Sarah Sheldon, Easwar Magesan, Jerry M. Chow and Jay M. Gambetta, "Procedure for systematic tuning up known cross-talk in the cross-resonance gate", PHYSICAL REVIEW A 93, 060302 (R) (2016) . Modified quantum Hamilton tomography replaces the echo-like cross-resonance pulse with a cross-resonance pulse.

So that the frequencies of the two qubits differ, they can be built differently. Alternatively or in addition, a magnetic field can be used to change the frequency of a qubit to arrive at a circuit with two qubits whose frequencies are different.

The first qubit to which the cross-resonance pulse is sent is called the control qubit. The other qubit is called the target qubit.

The first and second qubits may be superconducting qubits. The first qubit can be a transmon. The first qubit can be a CSFQ. The second qubit can be a transmon. The second qubit can be a CSFQ.

In one embodiment of the invention, both qubits are a transmon. The qubit with the higher frequency is selected as the control qubit. After applying a certain amplitude cross-resonance, the two-qubit state phase error is reduced. This increases the CR gate fidelity. By CR gate is meant the cross resonance gate.

In one embodiment of the invention, the control qubit is a CSFQ. The target qubit is a transmon. The circuit is built so that the frequency of the transmon is greater than the frequency of the CSFQ. Applying cross-resonance at a certain amplitude can improve CR gate fidelity (also called CR gate fidelity).

A control device for a qubit is preferably present, by means of which a qubit can be tuned. The frequency and the anharmonicity of the qubit can be changed by the control device. By being able to change the frequency of one qubit, a difference between the frequency and anharmonicity of the first qubit and the frequency of the second qubit can be optimized if necessary. Such optimization can improve fidelity. In one embodiment of the invention, a readout pulse is sent to the target qubit after the CR pulse has been applied to the control qubit for the duration of time t. The frequency of the readout pulse is preferably chosen such that the measured reflected pulse is minimal. The amplitude or power of the read-out pulse is preferably selected in such a way that the number of photons in the resonator, ie in the corresponding electrical conductor, is less than 1 on average. The resonator is an example of a coupler. It is a transmission line with a length equal to its natural frequency and consists of a superconductor that capacitively couples qubits. The number of photons in the resonator is proportional to the power of the readout pulse and the frequency. In practice, to ensure that the average number of photons is less than 1, ie in the single-photon range, the reflectance can be measured as a function of frequency at different microwave powers. It thereby shifts the resonant frequency at high power, commonly referred to as the bare resonator frequency, with decreasing microwave power (and thus also the number of photons) in the mean power to the lowest frequency and finally to the lowest resonant frequency when the system into the so-called "dressed state". At a "knee" just before reaching the "dressed state", the number of photons is usually of the order of 1. In practice, the microwave power is preferably set lower from this knee to ensure that one is really in a one -Photon area is located. For example, the microwave power can be set 10 dB to 30 dB lower, such as 20 dB. A state of the target qubit can be measured using the readout pulse.

In accordance with the present invention, by tuning the qubit parameters and the qubit-to-coupler and qubit-to-qubit capacitive coupling and the amplitude of the CR microwaves on the control qubit, the unwanted two-qubit phase error due to ZZ level repulsion can be suppressed and thus the CR Gate fidelity can be improved.

The qubits in the circuit can have the same anharmonicity sign. It is not necessary for the qubits of a circuit to have the same anharmonicity sign. The anharmonicity of qubits in a circuit can also be of opposite sign. Thus, one qubit of a circuit may be a transmon that has negative anharmonicity and another qubit may be a qubit of opposite sign, such as a CSFQ qubit. a qubit of a circuit can be a transmon and another qubit can be another transmon. A qubit of the circuit can be a CSFQ and another qubit can be another CSFQ.

Any single qubit gate is achieved by rotation in the Bloch sphere. The rotations between the different energy levels of a single qubit are induced by microwave pulses. Microwave pulses can be sent by a microwave generator to an antenna or to a transmission line coupled to the qubit. The frequency of the microwave pulses can be a resonant frequency in relation to the energy difference between two energy levels of a qubit. Individual qubits can be addressed over a dedicated transmission line, or over a common line if the other qubits are not in resonance. The axis of rotation can be tuned by quadrature amplitude modulation of the microwave pulse. The pulse length determines the angle of rotation.

The microwave that entangles two qubits is the cross-resonance gate. This cross-resonance gate, also known as a CR gate, is used to entangle qubits in a desired manner. The CR gate creates the desired ZX interaction that is used to create CNOT. If instead of a single CR pulse, a sequence of 4 pulses called "echo CR" is applied to the control qubit, some of the unwanted interactions like the X and Y rotation of the target qubit can be eliminated. Echo-CR preserves the desired ZX interaction and also cannot eliminate the two-qubit phase error introduced by the ZZ-repelling interaction.

The inventors have found that it is possible to eliminate unwanted phase errors in the two-qubit state in a circuit with two qubits, each interacting with a coupler and one of the qubits being driven by a cross-resonance pulse, by changing the parameters of the qubits and the coupling strength between qubit and coupler as well as the amplitude of a cross-resonance pulse can be tuned to each other. Qubits anharmonicities can have the same sign and qubits anharmonicities can have the opposite sign.

The invention is explained in more detail below with reference to figures. Show it Figure 1: circuit;

FIG. 2: pulse train;

FIG. 3: Circuit QED parameters for error-free transmon-transmon

Phase;

FIG. 4: Circuit QED parameters for error-free transmon-transmon

Phase;

FIG. 5: table;

FIG. 6: table;

FIG. 7: Diagram.

Figure 1 illustrates the basic structure with a first qubit 3, a second qubit 7 and a coupler 4 for indirect coupling of the two qubits 3 and 7 via the two coupling capacitors 8 and 9. The qubits 3 and 7 are also coupled directly across the capacitor 10. A first microwave transmission line 2 is coupled to the first qubit 3 . A second microwave transmission line 6 is coupled to the second qubit 7 . A first microwave port 1 is coupled to the first microwave transmission line 2 . A second microwave port 5 is coupled to the second microwave transmission line 6 .

The first qubit 3 can be provided as a target qubit. The second qubit 7 can be provided as a control qubit. A qubit 3, 7 can include superconducting conductor tracks. A qubit 3, 7 can include one or more Josephson junctions. The control qubit 7 can be a frequency tunable transmon. The control qubit 7 can also be a frequency tunable CSFQ. In Figure 1 we present an example circuit where the control qubit 7 is a frequency tunable transmon with two asymmetric Josephson junctions and the target qubit 3 is an example of a fixed frequency transmon with a Josephson junction.

The coupler 4 can be a bus resonator. The coupler 4 can be a superconductor, which is coupled to the two qubits 3 and 7 via a respective capacitance 8 and 9 . The first and second microwave ports 2 and 6 can be a superconductor, which can be coupled via capacitances to the associated qubits 3 and 7, respectively, and to the associated transmission line ports 1 and 5, respectively. Through the coupler 4 there is an indirect coupling between the two qubits 3 and 7.

The frequency of the first or the second qubit 3 or 7 can advantageously be tuned. The frequency of the control qubit can be adjusted in the case of FIG. For example, the tunable qubit can be tuned by a magnetic field penetrating the loop of two transitions in the asymmetric transmon. In this case, a controller can generate and change a magnetic field for tuning the qubit. The control device can include an electromagnet. The control qubit 3 can have a tunable frequency, e.g., an asymmetric transmon, and the target qubit can be a fixed-frequency transmon.

The second qubit 7 can be coupled to a readout device. The readout device can include a microwave generator for generating a readout pulse.

FIG. 2 schematically shows the transmission of a pulse sequence to the control qubit 7. The pulse height is plotted on the y-axis over time t on the x-axis. The control qubit 7 and the target qubit 3 are set to be in the ground state |00>. This is referred to as "state preparation". A cross-resonance pulse 11 with a set amplitude and for the duration of time t is applied to the resonator 6 via the connection 5 and sent from there to the control qubit 7 . This is referred to as "CR drive". After the emission of the cross-resonance pulse 11, the repulsion of the qubit level should be measured. This is referred to as "target state tomography". The target state tomography step is described in the publication Sarah Sheldon, Easwar Magesan, Jerry M. Chow and Jay M. Gambetta, "Procedure for systematic tuning up known cross-talk in the cross-resonance gate", PHYSICAL REVIEW A 93, 060302 (R) (2016). For the target state tomography step, we send the microwave pulse 13 to the port 1 , then it travels to the target qubit 3 via the resonator 2. There are three types of the microwave pulse 13. The first type of the microwave pulse 13 rotates the target qubit 3 around the Angle n/2 along the X axis of the Bloch sphere. The second type of microwave pulse 13 rotates the target qubit 3 by the angle π/2 along the Y-axis of the Bloch sphere. The third type of microwave pulse 13 rotates the target qubit 3 by the angle π/2 along the Z-axis of the Bloch sphere. We apply only one of the three types of microwaves 13 to the target qubit and then measure the target qubit state in 14. After the measurement, we reinitialize the state in the state preparation step, apply an unchanged CR drive pulse with the same amplitude and time length t, then one of the three microwave types 13 and perform the measurement again. We repeat this 13 thousand times with one of the three types of microwave. This determines the average probability of the target qubit state projected onto the x and y and z axes. We plot the mean state probability along the x-axis by <x>, we plot the mean state probability along the y-axis by <y>, and we plot the mean state probability along the z-axis by <z>. The target qubit state tomography characterizes the state of the target qubit by three numbers <x>, <y>, <z>. After determining <x>, <y>, and <z> associated with CR length t and an amplitude, we change the CR gate length t and keep the amplitude the same. Then we repeat the target quantum state tomography and determine the new projected target state components <x>, <y> and <z>. In this way we find the <x>(t) , <y>(t) and <z>(t) , which depend on the CR pulse length.

We reinitialize the two qubits in the |0> state and this time we apply an X rotation gate by angle n to the control qubit each time after the initialization step. This can be done by applying the microwave pulse 12 to the control qubit. In this way, the control qubit is always initialized in the |1> state, while the target qubit is in the |0> state. The process of applying CR driver stepping and target state tomography is similarly repeated. The process of determining <x>(t), <y>(t), and <z>(t) is repeated for the case that control qubit 7 was initialized in state |1>.

A Hamiltonian model is used to determine the same target state projections <x>(t), <y>(t), and <z>(t) that will be control state dependent. As in Sarah Sheldon, et al. PHYSICAL REVIEW A 93, 060302 (R) (2016), a ZZ- interaction term can be included in the Hamiltonian model. This ZZ interaction term corresponds to the coupling strength y of a two-qubit state phase error in the presence of a CR gate.

The repetition of the steps of the quantum Hamilton tomography of FIG. 2 with a different amplitude for the CR pulse 11 will produce a different y and thus a different two Determine qubit phase error. Repeating the same experiment with a specific amplitude of the CR pulse 11 sets y=0 and therefore provides no two-qubit state phase error.

The frequency of the two cross-resonance pulses corresponds to the frequency of the target qubit 3.

Two microwave generators can be provided to generate the CR pulses. A first microwave generator generates the TT rotation along the X-axis pulses 12. A second microwave generator generates the cross-resonance pulses 1. An adder 15 can be provided in order to send the pulse sequence via the microwave connection 5 to the first -qubit 7. A third microwave generator can be provided for sending a read-out pulse. A readout pulse can be sent to the second qubit 7 by the third microwave generator via the second microwave port 5 to generate either of the two types of X and Y rotations by TT/2 on the target qubit 3 by the microwave pulse 13 . For rotation along the Z axis, we need two microwave generators instead of one to generate an X(TT/2) and a Y(TT/2). In one performance, the microwave pulse 13 is formed from two consecutive pulses, first an X rotation through TT/2, followed by a Y rotation through TT/2. After reinitialization, the pulse 13 will this time first perform a Y rotation through TT/2, followed by an X rotation through TT/2. The Z rotation through the angle TT/2 is the result of the difference of the results measured with the opposite orders. A fifth microwave generator can be provided for the transmission of a read-out pulse 15 . A readout pulse can be sent from the third microwave generator to the qubit 3 via the second microwave connection 1 .

The two-qubit phase error y from the CR pulse depends on the CR amplitude and the frequency match between the controller and the target qubit. The relationship is Y=g+n(A)Q ² , where g is the idle two-qubit error and Q is the amplitude of the CR pulse, and q(A) is a function of the frequency detuning A=w _t arget- control Eliminating the two-qubit phase error using the CR pulse means we set y=0. This means that for a circuit with a specific static error g and detuning frequency A, elimination occurs at a specific amplitude Q. Figure 3 relates to the theoretical results of circuit QED modeling of a circuit in which both the control and target qubits are transmon. The control qubit 7 is driven by a CR pulse 11 with amplitude 12. The frequency of the control qubit is o>c and the frequency of the target qubit is t t. For the control and target qubits that have the same value of anharmonicity, the control qubit has a greater frequency. The difference between the frequency of the target qubit and the frequency of the control qubit is the frequency of the transmon-transmon detuning. The detuning frequency A can be negative. We show the detuning frequency on the x-axis of Figure 3 and the amplitude of the CR pulse on the y-axis. Rectangles and solid line show the CR pulse amplitude estimates at which, for any detuning frequency A, the repulsion level between E11 and non-calculation states is set to zero. The solid line are the solutions taken from perturbation theory. The rectangles show the results of the exact solution.

FIG 4 relates to the theoretical results of circuit QED modeling of a circuit where the control qubit is a CSFQ and the target qubit is a transmon. The control qubit 7 is driven by a CR pulse 11 of amplitude Q. The frequency of the control qubit is o>c and the frequency of the target qubit is tt. In the control qubit the anharmonicity is positive and in the target qubit the anharmonicity is negative. The anharmonicity of the control qubit can be greater than the absolute value of the anharmonicity in the target qubits. In this case, the frequency of the control qubit is less than the frequency of the target qubit. The difference between the frequency of the target qubit and the frequency of the control qubit is the CSFQ transmon detune frequency. The detuning frequency A can be positive. We show the detuning frequency on the x-axis of FIG. 4 and the amplitude of the CR pulse on the y-axis. Rectangles and solid line show the estimated amplitude of the CR pulse at which, for each detuning frequency A, the qubit level repulsion vanishes. The solid line shows the result from the perturbation theory. The rectangles show that the results are not disruptive and give more precise results.

In order to be able to determine experimentally whether the state dephasing due to level repulsion is zero or at least close to zero, Hamilton tomography is required for the determination. The Hamiltonian tomography can be found in the publication Sarah Sheldon, Easwar Magesan, Jerry M Chow and Jay M Gambetta, "Procedure for systematic tuning up cross-talk in the cross-resonance gate", PHYSICAL REVIEW A 93, 060302 (R) (2016). Known methods can therefore be used. A cross-resonance drive is applied for some time and the Rabi vibrations are measured on the target qubit. We project the state of the target qubit onto x, y, and z after the Rabi drive, and repeat for the control qubit in | 0) and | 1). In this way we find the exact interaction strengths of each of the above terms in the CR-Hamiltonian. This is called a CR tomography experiment.

In the first step, the two qubits are initialized in the state |00>. A CR pulse is sent to control qubit 7.

The dephasing of the state from the repulsion plane is then measured by CR tomography. If the value is non-zero, the amplitude of the cross-resonance pulse is changed and the process repeated. If the value is equal to zero, the optimal amplitude sought has been found.

The results shown in Figure 5 were found for 10 different cases. The first five cases show results for the previously described case where the first qubit is a CSFQ and the second qubit is a transmon. As described above, the following five cases relate to a circuit in which the first qubit and the second qubit are a transmon. In all cases, the two qubits were entangled. The table shows that it is not always possible to find a value of zero. In these cases, the amplitude closest to zero is selected.

Figure 6 shows the result of applying two qubit gates CNOT to two pairs of qubits. The gate CNOT acts on the qubits for the duration of time t. The two-qubit phase error is present in the first pair. The phase error is proportional to ±yt. The sign depends on the state of the two qubits. The sign is positive when the two qubits have the same states. The sign is negative if the states of the qubits are different. In the second pair, by coordinating the qubit parameters and the amplitude of a microwave pulse, we eliminate the fundamental two-qubit phase error.

Figure 7 shows the value of the two-qubit phase error y as a function of the CR pulse amplitude in two different transmon-transmon circuits 16 and 17. In the circuit 16, the phase error initially decreases by increasing the amplitude, but begins to rise after reaching a positive minimum without crossing zero. Therefore, it is impossible to make the circuit 16 two-qubit phase error free. In the circuit 17, by increasing the amplitude of the CR pulse, the phase error decreases and crosses the zero and changes sign. The point at which the zero crossing occurs is the specific amplitude that eliminates the qubit-two-qubit phase error.

Claims

patent claims

English Method of operating a circuit having a first qubit (7) and a second qubit (3), the circuit being arranged such that the frequency of the first qubit (7) differs from the frequency of the second qubit (3). , with a coupler (4) which couples the first qubit (7) and the second qubit (3), wherein a cross-resonance pulse is sent to the first qubit (7), the amplitude of the cross-resonance pulse being selected such that the two- Qubit phase error is minimal, or at least substantially minimal, with a two-qubit phase error made by repulsion between qubit energy levels and non-computational levels.

2. The method according to claim 1, characterized in that the amplitude of the cross-resonance pulse is selected so that the two-qubit phase error is set to zero.

3. The method according to any one of the preceding claims, characterized in that a control device for a qubit (7) is present, through which the frequency of the qubit can be tuned.

4. The method according to the preceding claim, characterized in that the control device can generate and change a magnetic field.

5. The method according to the preceding claim, characterized in that the control device comprises an electromagnet.

6. The method according to any one of the preceding claims, characterized in that the frequency of the cross-resonance pulse corresponds to the frequency of the second qubit (3).

7. The method according to any one of the preceding claims, characterized in that the first qubit (7) is a transmon and the second qubit (3) is a transmon.

8. The method according to the preceding claim, characterized in that the frequency of the control qubit (7) is greater than the frequency of the target qubit (3).

9. The method according to any one of the preceding claims 1 to 7, characterized in that the control qubit (7) is a CSFQ and the target - qubit (7) is a transmon.

10. The method according to the preceding claim, characterized in that the frequency of the control - qubit (7) is smaller than the frequency of the target - qubit (3).

11. The method according to any one of the preceding claims, characterized in that the target qubit (3) is coupled to a readout device.