METHOD AND ARRANGEMENT FOR READING OUT THE STATE OF A QUBIT

Abstract

For reading out a state of a qubit, a readout input waveform is injected into a system that comprises an information storage element for storing the state of the qubit and a readout resonator that is electromagnetically coupled to said information storage element. A readout output waveform is extracted from said system and detected. The injection of the readout input waveform takes place through an excitation port that is also used to inject excitation waveforms to the information storage element for affecting the state of the qubit. A phase of the readout input waveform is controllably shifted in the course of injecting it into the system.

Claims

1: An arrangement for reading out a state of a qubit, comprising: an information storage element for storing the state of the qubit; a readout resonator electromagnetically coupled to said information storage element; an excitation port for injecting excitation waveforms to the information storage element for affecting the state of the qubit; one or more readout ports for injecting readout input waveforms to the system comprising said information storage element and said readout resonator, and for extracting readout output waveforms from the system; a readout waveform source for generating said readout input waveforms; and a readout waveform detector for detecting said readout output waveforms, wherein said readout waveform source is configured to inject said readout input waveforms into the system simultaneously both through said excitation port and through a first readout port of said one or more readout ports, said first readout port being different than said excitation port, and wherein said readout waveform source is configured to controllably shift a phase of a readout input waveform in the course of injecting it into the system.

2: The arrangement according to claim 1, wherein: said excitation port is coupled to said information storage element, and one or more of said readout ports are coupled to said resonator.

3: The arrangement according to claim 2, wherein said information storage element and said readout resonator are made of superconductor materials.

4: The arrangement according to claim 3, wherein said information storage element is a transmon.

5: The arrangement according to claim 1, wherein: said readout waveform source is configured to control the phase and amplitude of both the readout input waveform injected into the system through said excitation port and the readout input waveform injected into the system through said first readout port.

6: The arrangement according to claim 5, wherein said readout waveform source is configured to: inject into the system a first pair of simultaneous readout input waveforms through said excitation port and said first readout port respectively, with phases and amplitudes of said first pair of readout input waveforms matched in order to maintain a mean point of a first probability distribution at the origin of an I-Q space while moving a mean point of a second probability distribution away from said origin of said I-Q space, said first probability distribution being associated with a first possible state of a qubit stored in said information storage element and said second probability distribution being associated with a second possible state of the qubit stored in said information storage element; and subsequently inject into the system a second pair of simultaneous readout input waveforms through said excitation port and said first readout port respectively, with phases and amplitudes of said second pair of readout input waveforms matched in order to move the mean point of said second probability distribution back to the origin of the I-Q space.

7: The arrangement according to claim 6, wherein said readout waveform detector is configured to perform a detection of a readout output waveform extracted from the system before said subsequent injection into the system of the second pair of simultaneous readout input waveforms.

8: A method for reading out a state of a qubit, comprising: injecting a readout input waveform into a system that comprises an information storage element for storing the state of the qubit and a readout resonator that is electromagnetically coupled to said information storage element; and detecting a readout output waveform extracted from said system, wherein said injecting of the readout input waveform takes place through an excitation port that is also used to inject excitation waveforms to the information storage element for affecting the state of the qubit, and simultaneously through a first readout port that is different than said excitation port, and wherein a phase of the readout input waveform is controllably shifted in the course of injecting it into the system.

9: The method according to claim 8, further comprising controlling the phase and amplitude of both the readout input waveform injected into the system through said excitation port and the readout input waveform injected into the system through said first readout port.

10: The method according to claim 9, further comprising: injecting into the system a first pair of simultaneous readout input waveforms through said excitation port and said first readout port respectively, with phases and amplitudes of said first pair of readout input waveforms matched in order to maintain a mean point of a first probability distribution at the origin of an I-Q space while moving a mean point of a second probability distribution away from said origin of said I-Q space, said first probability distribution being associated with a first possible state of a qubit stored in said information storage element and said second probability distribution being associated with a second possible state of the qubit stored in said information storage element; and subsequently injecting into the system a second pair of simultaneous readout input waveforms through said excitation port and said first readout port respectively, with phases and amplitudes of said second pair of readout input waveforms matched in order to move the mean point of said second probability distribution back to the origin of the I-Q space.

11: The method according to claim 10, further comprising: detecting a readout output waveform extracted from the system before said subsequent injection into the system of the second pair of simultaneous readout input waveforms.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The accompanying drawings, which are included to provide a further understanding of the invention and constitute a part of this specification, illustrate embodiments of the invention and together with the description help to explain the principles of the invention. In the drawings:

[0026] FIG. 1 illustrates a superconductive quantum memory circuit,

[0027] FIG. 2 illustrates an equivalent circuit diagram,

[0028] FIG. 3 illustrates the separation of state-associated probability distributions in a first case,

[0029] FIG. 4 illustrates a system with a qubit and a readout resonator,

[0030] FIG. 5 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0031] FIG. 6 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0032] FIG. 7 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0033] FIG. 8 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0034] FIG. 9 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0035] FIG. 10 illustrates a number of mathematical formulae explaining embodiments of the invention,

[0036] FIG. 11 illustrates an arrangement for reading out the state of a qubit,

[0037] FIG. 12 illustrates the separation of state-associated probability distributions in a second case,

[0038] FIG. 13 illustrates the separation of state-associated probability distributions in a third case,

[0039] FIG. 14 illustrates the separation of state-associated probability distributions in a fourth case,

[0040] FIG. 15 illustrates a comparison of the separation of state-associated probability distributions in a number of cases,

[0041] FIG. 16 illustrates steps of a method for reading out the state of a qubit,

[0042] FIG. 17 illustrates steps of a method for reading out the state of a qubit, and

[0043] FIG. 18 illustrates steps of a method for reading out the state of a qubit.

DETAILED DESCRIPTION

[0044] FIG. 4 is a schematic illustration of the principle of an arrangement for reading out the state of a qubit. The arrangement comprises an information storage element 101 for storing the state of qubit. The two horizontal lines in the drawing mark the two possible basis states that a qubit may have. In the technical field of hardware devices for quantum computing it is customary to use the term qubit not only for the conceptual basic unit of information but also for the piece of hardware that constitutes the information storage element 101.

[0045] The arrangement comprises also a readout resonator 102 that is electromagnetically coupled to the information storage element or qubit 101. The readout resonator 102 is a harmonic oscillator and it has a certain resonance frequency. The strength of the electromagnetic coupling between the resonator 102 and the information storage element (or qubit) 101 can be described with a coupling coefficient g. For the ease of reference, the qubit 101 and its readout resonator 102 can be commonly referred to as “the system”.

[0046] The arrangement comprises an excitation port 103 for injecting excitation waveforms 401 to the information storage element 101. The excitation waveforms affect the state of the qubit in the known way. In the general parlance of the technical field it is common to speak about “exciting” the qubit, which is essentially synonymous with injecting excitation waveforms through the excitation port 103.

[0047] The arrangement comprises one or more readout ports 104 for injecting readout input waveforms 402 to the system. The one or more readout ports 104 are also used for extracting readout output waveforms 403 from the system. Injecting readout input waveforms 402 to the system is generally referred to as driving the resonator 102. A coupling coefficient κ (smallcase kappa in Greek letters) describes the characteristic decay time from the resonator 102 to the readout port(s) 104. The relative magnitudes of the constants g and κ have certain significance to the ways in which the readout mechanism operates, as will be described in more detail later in this text.

[0048] The readout input waveforms 402 originate from a readout waveform source, which is not shown in FIG. 4. Detection of readout output waveforms that are extracted from the system takes place in a readout waveform detector, which is also not shown in FIG. 4.

[0049] In contrast to what has been conventional in the technical field, the readout waveform input source may be arranged to inject at least some of the readout input waveforms into the system through at least the excitation port 103. Thus in a way the excitation port 103 becomes simultaneously one of the readout ports of the system. This has a significant effect on the speed at which reading out the state of the qubit can proceed.

[0050] Conceptually the situation can be explained as follows. In the conventional readout scheme, in which readout input waveforms were injected solely through the readout port 104, the resonator 102 was empty to begin with. The readout input waveforms, or readout photons as they are also called, had to first populate the resonator 102 before they could begin interacting with the state stored in the qubit 101. The useful information gained from the output waveform is proportional to the product of an amplitude and a phase, so only after the amplitude of the oscillations in the resonator 102 reached a meaningful magnitude and had sufficient time to interact with the state in the qubit 101 through the coupling g it became reasonable to detect their phase.

[0051] When a readout input waveform is injected into the system through the excitation port 103, it “meets” immediately the state that is stored in the qubit 101 and can thus begin interacting with it already before it ends up in the resonator 102. In other words, the resonator 102 begins to get populated with readout photons the phase of which already reflects the state of the qubit that is to be read out. As a result it becomes possible to detect appropriate readout output waveforms earlier than in the conventional method.

[0052] A more formal treatment of the situation is as follows. Let the eigenfrequencies of the uncoupled qubit 101 be ω.sub.k=kω.sub.r+Δ.sub.k, where ω.sub.r is the resonance frequency of the resonator 102 and Δ.sub.k denotes the detuning between the k:th energy levels of the qubit and the resonator. Δ.sub.0=0 for the ground state, Δ.sub.1=Δ, for the first excited state, Δ.sub.2=2Δ+α for the second excited state where a is the anharmonicity, and so on. In the dispersive regime the detuning is larger than the qubit-resonator coupling g, which means that |Δ|>>g. The Hamiltonian that describes the system can be written as illustrated on line (1) of FIG. 5, and the free, interaction, qubit-driving, and resonator-driving Hamiltonians are respectively given by lines (2), (3), (4), and (5) in FIG. 5.

[0053] In the mathematical notation used â denotes the annihilation operator of the resonator mode, and |k) refers to the k:th eigenstate of the qubit. All subscripts “r” refer to the resonator, subscripts “q” to the qubit, and subscripts “d” to the readout (i.e. driving) waveform.

[0054] For a transmon, the coupling constants for different transmon levels are typically assumed to be of the form g.sub.k=g√{square root over (k+1)}, λ.sub.k=√{square root over (k+1)}. The real driving (i.e. readout) waveforms {tilde over (Ω)}.sub.r/q(t) at driving frequency ω.sub.d are constructed from the real and imaginary parts (i.e. I and Q quadratures) of the complex amplitudes as shown on line (6) of FIG. 5.

[0055] The Hamiltonian Ĥ.sub.total can be transformed into the frame rotating at the angular frequency ω.sub.d. Applying the unitary operator .Math..sub.1 given on line (7) of FIG. 5 and employing the rotating wave approximations justified by g<<|2ω.sub.r| and |ω.sub.r−ω.sub.d|<<|ω.sub.r+ω.sub.dr| gives the transformed Hamiltonians given on lines (8), (9), (10), and (11) of FIG. 6. The acronym H.c. is used to denote the Hermitian conjugate, and {tilde over (Δ)}.sub.k=Δ.sub.k+kω.sub.r−kω.sub.d denotes the shifted de-tunings.

[0056] Ignoring Ĥ.sub.RD′ for a moment, the total transformed Hamiltonian Ĥ.sub.total′ is given by line 12 in FIG. 7.

[0057] In the conventional readout scheme, in which readout waveforms are only injected into the system through the readout port, Ω.sub.d=0. Thus the phase space distribution of the resonator will rotate about the origin at an angular frequency that depends on the state of the qubit. Here we make the key observation that the frame is displaced by α.sub.VO≡−Ω.sub.qλ.sub.k/g.sub.k=−Ω.sub.q/g. Thus, in the non-shifted frame, the phase space distribution of the resonator should rotate about the point α.sub.VO. The location of α.sub.VO is fully controllable by the readout waveform, Ω.sub.qd and ω.sub.d.

[0058] To account for the decay of the resonator state, we use the Lindblad master equation given on line (13) of FIG. 7, where ρ is the reduced density operator of the resonator, κ denotes the resonator energy decay rate, and [â]ρ=âρâ.sup.†−½(â.sup.†âρ+ρâ.sup.†â).

[0059] To make this observation more evident, we perform the standard dispersive approximation. We begin by making another transformation using the operator .Math..sub.2 given by line (14) in FIG. 7. We compute Ĥ.sub.i″=.Math..sub.2Ĥ.sub.i′.Math..sub.2.sup.† up to the second order in g.sub.k/{tilde over (Δ)}.sub.k+1 under the assumption g.sub.k<<{tilde over (Δ)}.sub.k+1 for ∀k. For clarity we may restrict the formulas to only the first three levels of the transmon ({|g,|e,|f}={|0,|1,|2}). The Hamiltonians assume the forms given on lines (15), (16), and (17) of FIG. 7. Here we have defined the dispersive constants χ.sub.0=g.sub.0.sup.2/{tilde over (Δ)}.sub.1 and λ.sub.1=g.sub.1.sup.2/({tilde over (Δ)}.sub.2−{tilde over (Δ)}.sub.1), and Ω.sub.rd′=(≠.sub.rd/2) exp(it(ω.sub.r−ω.sub.d)). Finally, introducing the displaced operator {circumflex over (b)}=â−α.sub.VO, the total Hamiltonian assumes the form given on lines (18) to (21) of FIG. 8.

[0060] Line (18) describes the constant frequency shifts caused by the coupling and the driving. Line (19) shows that driving from the qubit side, i.e. injecting readout waveforms through the excitation port into the system, tilts the qubit Hamiltonian. Line (20) is important to the readout scheme considered here, because it predicts that any coherent state will rotate about point α.sub.VO. The angular frequencies of these rotations may be set to be equal to +χ≡χ.sub.1/2−χ.sub.0 and −χ for α.sub.g and α.sub.e respectively, by choosing ω.sub.r−ω.sub.d=χ.sub.1/2. Line (21) in FIG. 8 shows that the transformation has an effect on the amplitude of the resonator drive that may be compensated by changing Ω.sub.r. The Hamiltonian of a conventional dispersive system is obtained by setting α.sub.VO=0.

[0061] Using the equation on line (13) of FIG. 7 with the approximate Hamiltonian Ĥ.sub.total′ we obtain an analytical equation for the expectation value α.sub.j=â.sub.j, j∈{g,e} as line (22) in FIG. 9. Assuming that the resonator is empty to start with (so-called “vacuum state”) and that the readout pulses do not change, ∂α.sub.VO/∂t=0, the solution is given on line (23) of FIG. 9.

[0062] The formal treatment given above is valid for a general case, and it is not bound to e.g. any particular physical implementation of the qubit. The following three special cases can be noted.

[0063] The first special case is a conventional readout scheme in which no readout waveforms are injected to the system through the excitation port, meaning that Ω.sub.qd=0. In that case the probability distributions associated with the two qubit states |g and |e will rotate around different points in the phase space. They will initially advance in the same direction, as was described above in association with the trajectories shown in FIG. 3. The state separation at t<χ.sup.−1 increases quadratically in time, as shown by line (24) in FIG. 10.

[0064] The second special case is a case in which readout waveforms are injected to the system only through the excitation port, meaning that Ω.sub.rd=0. This readout scheme may be called the back door readout scheme to illustrate its difference to the conventional alternative. The probability distributions associated with the two qubit states |g and |e will rotate around point z, but at different frequencies. The state separation at t<χ.sup.−1 increases linearly in time, as shown by line (25) in FIG. 10. Further evolution depends on the magnitude of κ. For small κ, the probability distributions will rotate around z while converging slowly to their respective steady states. This may make single shot readout more challenging as S.sub.back will oscillate. The same possible problem exists also in the conventional readout scheme, but it can be avoided by having a large κ.

[0065] The third special case is to inject readout waveforms into the system through both the excitation and readout ports, in such a way that the numerator in Equation (23) equals zero in FIG. 9. This will cause the probability distribution associated with the qubit state |g to remain at the origin. The probability distribution associated with the qubit state |e will rotate around point α(1−χ.sub.|g/χ.sub.|e)). The state separation at t<χ.sup.−1 increases as in the previous case, i.e. as S.sub.back(t).

[0066] FIG. 11 illustrate an arrangement for reading out the state of a qubit 101, in which the equivalent circuit diagram representation of FIG. 2 is adopted for illustrative comparison. The arrangement comprises a readout waveform source 1101 for generating readout input waveforms. The arrangement comprises also a readout waveform detector 1102 for detecting readout output waveforms extracted from the system that comprise the qubit (or information storage element) 101 and the readout resonator 102. The readout waveform source 1101 is arranged to inject readout input waveforms into the system through at least the excitation port 103. In the embodiment shown in FIG. 11 the readout waveform source 1101 is additionally arranged to inject readout input waveforms into the system through the readout port 206.

[0067] The readout waveform source 1101 is arranged to controllably shift the phases of readout input waveforms in the course of injecting them into the system.

[0068] This capability is schematically illustrated in FIG. 11 with the controllable phase shifters 1103 and 1104 in the lines leading from the output of the readout waveform source 1101 to the readout and excitation ports 206 and 103 respectively. The readout waveform source 1101 may also be arranged to control the amplitude of the readout input waveforms. This capability of the readout waveform source 1101 is schematically illustrated in FIG. 11 with the controllable attenuators 1105 and 1106 in the lines leading from the output of the readout waveform source 1101 to the readout and excitation ports 206 and 103 respectively.

[0069] As shown in FIG. 11, the excitation port 103 is coupled to the qubit (or information storage element) 101, and the readout port 206 is coupled to the resonator 102. To be quite exact, since the excitation port 103 doubles as a readout port (because readout input waveforms are injected therethrough) in FIG. 11, it may be said that one other readout port 206 that is different than the excitation port 103 is coupled to the resonator 102.

[0070] The qubit (or information storage element) 101 and the resonator 102 can be made of superconductor materials: as an example, they may appear on a superconductive quantum memory circuit like that shown in FIG. 1. However, this is not an essential requirement, and other kinds of qubit technologies could be used. A superconductor material means here a material that can be made superconductive by cooling it to a sufficiently low temperature. An example of such materials is aluminum, but also other superconductor materials like molybdenum, niobium, tin, tantalum, or lead can be used.

[0071] For operation, a superconductive quantum memory circuit is cooled to a very low temperature, which can be some kelvins, or well under one kelvin, or in the order of some tens of millikelvins. The qubit 101 is preferably an anharmonic oscillator, such as a transmon.

[0072] FIG. 12 shows an example of how the mean points of the two probability distributions associated with the two qubit states |g and |e may move in the phase space in the “back door readout” case. This refers to the case in which the readout waveform source 1101 injects readout waveforms to the system only through the excitation port 103. The graphical notation in FIG. 12 is the same as in FIG. 3 earlier.

[0073] FIG. 13 shows an example of how the mean points of the two probability distributions associated with the two qubit states |g and |e may move in the phase space in the “asymmetric back door readout” case. This refers to the case in which the readout waveform source 1101 injects readout waveforms to the system simultaneously both through the excitation port 103 and through the readout port 206. For generality it may be said that the arrangement may comprise one or more readout ports, and port 206 is a first readout port of said one or more readout ports and different than the excitation port 103.

[0074] In particular, in the case of FIG. 13 the readout waveform source 1101 may be configured to control the phase and amplitude of both the readout waveform injected into the system through the excitation port 103 and the readout waveform injected into the system through the first readout port 206, so that the numerator equals zero in Equation (23) in FIG. 9. This makes one of the probability distributions remain at or very close to the origin of the I-Q space, while the other moves away from it along a curved trajectory.

[0075] FIG. 14 illustrates how the principles considered above can be employed to perform a fast reset of the readout resonator. First, the readout waveform source 1101 injects into the system a first pair of simultaneous readout waveforms through the excitation port 103 and the readout port 206 respectively. The phases and amplitudes of this first pair of readout waveforms are matched following the principle explained above with reference to FIG. 13: the matching is made in order to maintain a mean point of a first probability distribution at the origin of an I-Q space while moving a mean point of a second probability distribution away from the origin of the I-Q space (arrow 1401). Here the first probability distribution is associated with a first possible state of a qubit stored in the information storage element 101, and the second probability distribution is associated with a second possible state of the qubit stored in the information storage element 101.

[0076] Subsequently the readout waveform source 1101 injects into the system a second pair of simultaneous readout waveforms through said excitation port and said first readout port respectively. The phases and amplitudes of this second pair of readout waveforms are matched in order to move the mean point of said second probability distribution back to the origin of the I-Q space (arrow 1402 in FIG. 14). This latter step is performed instead of just waiting that the photons of the readout waveform would naturally decay from the readout resonator, which would eventually bring the two probability distributions to the same point again in the I-Q space.

[0077] Not only the resetting of the resonator but also the detection of a readout waveform extracted from the system can take place faster than in the conventional method. The readout waveform detector 1102 may be configured to perform a detection before the latter step described above, i.e. before the readout waveform source 1101 injects the second pair of simultaneous readout waveforms to the system. Due to the linear increase in the state separation, a shorter integration time in detection gives sufficiently reliable results. If a slightly different viewpoint is taken, the detection result can be made more reliable if the same integration time is used as in the prior art method.

[0078] FIG. 15 shows a graphical comparison of how the state separation develops over time in the conventional (graph 1501), back door (graph 1502), and asymmetric back door (graph 1503) readout schemes.

[0079] FIG. 16 is a basic schematic illustration of a method for reading out the state of a qubit. Step 1601 represents the presumption that something may have changed the state of the qubit since it was read out last time, because otherwise there would be little reason to read it out again. Step 1602 comprises injecting a readout input waveform into a system that comprises an information storage element for storing the state of the qubit and a readout resonator that is electromagnetically coupled to said information storage element. Step 1603 comprises detecting a readout output waveform extracted from said system. In accordance with what was described above with reference to the arrangement, the injecting of the readout input waveform in step 1602 takes place through an excitation port that is also used to inject excitation waveforms to the information storage element for affecting the state of the qubit. There is also a step 1604 showing how the phase of the readout input waveform is controllably shifted in the course of injecting it into the system.

[0080] FIG. 17 shows how the step 1602 of injecting a readout input waveform into the system may comprise injecting readout waveforms into the system simultaneously both through said excitation port (substep 1701) and through a first readout port of said system (substep 1702). For clarity it can be emphasized that here said first readout port is different than said excitation port. FIG. 17 also shows how the step 1604 of controllably shifting the phase of a readout input waveform may involve controlling the phase and amplitude of both the readout waveform injected into the system through said excitation port and the readout waveform injected into the system through said first readout port.

[0081] FIG. 18 corresponds to what was previously explained with reference to FIG. 14. The method of FIG. 18 comprises injecting into the system a first pair of simultaneous readout waveforms through said excitation port and said first readout port respectively at step 1801. The phases and amplitudes of said first pair of readout waveforms may be matched according to step 1802, in order to maintain a mean point of a first probability distribution at the origin of an I-Q space while moving a mean point of a second probability distribution away from said origin of said I-Q space. Here said first probability distribution is associated with a first possible state of a qubit stored in said information storage element, and said second probability distribution is associated with a second possible state of the qubit stored in said information storage element.

[0082] The method of FIG. 18 comprises subsequently injecting into the system a second pair of simultaneous readout waveforms through said excitation port and said first readout port respectively at step 1803. The phases and amplitudes of said second pair of readout waveforms may be matched in accordance with step 1804 in order to move the mean point of said second probability distribution back to the origin of the I-Q space. The detecting of a readout output waveform extracted from the system at step 1603 may take place before said subsequent injection into the system of the second pair of simultaneous readout waveforms at step 1803.

[0083] It is obvious to a person skilled in the art that with the advancement of technology, the basic idea of the invention may be implemented in various ways. The invention and its embodiments are thus not limited to the examples described above, instead they may vary within the scope of the claims.