Patent Publication Number: US-11379749-B2

Title: Identifying noise sources in quantum systems via multi-level noise spectroscopy

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     This invention was made with Government support under Grant No. W911NF-14-1-0682 awarded by the Army Research Office (ARO). The Government has certain rights in the invention. 
    
    
     BACKGROUND 
     System noise identification is crucial to the engineering of robust quantum systems, relevant in quantum information processing and quantum sensing. Since noise in quantum devices places serious limitations on what quantum devices can achieve in the near term, identifying dominant sources of noise and eliminating them is crucial for improving future quantum devices. 
     Quantum noise spectroscopy (QNS) protocols have been developed and implemented to characterize noise processes in quantum systems. Existing approaches aim to characterize the spectral properties of environmental noise using a qubit as a spectrometer. A qubit spectrometer can be approximated as a two-level system, if higher levels of a qubit are not affected while coupling the two fundamental qubit levels to an electromagnetic field. By driving the qubit with suitably tailored external microwave control fields and measuring its response in the presence of environmental noise, the spectral content of the noise can be extracted. QNS protocols have been explored for both pulsed (free-evolution) and continuous (driven-evolution) control schemes and experimentally implemented across many qubit platforms—including nitrogen vacancy centers in diamond, nuclear spins, semiconductor quantum dots, trapped ions, and superconducting quantum circuits. 
     Although existing quantum noise spectroscopy (QNS) protocols measure an aggregate amount of noise affecting a quantum system, they generally cannot distinguish between the underlying processes that contribute to it. 
     Among existing QNS protocols, the spin-locking approach has been shown to be applicable to both classical and non-classical noise spectra. It uses a relatively straightforward relaxometry analysis to extract the spectral decomposition of the environmental noise affecting single qubits, and it has recently been extended to measure the cross-spectra of spatially correlated noise in multi-qubit systems. However, despite its efficacy, like most QNS protocols today, it was used and experimentally demonstrated only in the two-level approximation. 
     SUMMARY 
     Disclosed herein is a QNS protocol based on the spin-locking technique that is applicable to multi-level systems and allows for extracting the noise contributions from various noise sources, along with embodiments of systems and methods using the QNS protocol. Disclosed embodiments have been experimentally demonstrated using a flux-tunable transmon qubit, acting as a multi-level noise sensor. 
     It is appreciated herein that existing QNS protocols developed within a two-level system approximation have certain limitations (for example, limited bandwidth) when applied to weakly anharmonic qubits such as the transmon, the gatemon, or the capacitively shunted flux qubit, which are among the most promising candidates for realizing scalable quantum information processors. Since the presently disclosed noise spectroscopy technique incorporates the effects of higher-excited states, it is readily applicable in these weakly anharmonic superconducting qubits and is therefore compatible with state-of-the-art superconducting circuit implementations. 
     According to one aspect of the present disclosure, a method for identifying and discriminating contributions from one or more noise sources using the multi-level structure of a quantum system with three or more levels can include: preparing the quantum system in a predetermined state; applying one or more control signals to the quantum system; measuring values of one or more observables of the quantum system that quantify the quantum system&#39;s response to the noise sources and the one or more applied control signals; extracting noise spectra information associated with the noise sources from the measured values; and identifying contributions from the one or more noise sources based on the noise spectra information. 
     In some embodiments, the method can include improving performance of the quantum system based on the extracted noise spectra information and the identified contributions from the one or more noise sources. In some embodiments, the quantum system may include a plurality of superconducting qubits. In some embodiments, the superconducting qubits can include transmon or gatemon qubits. In some embodiments, the quantum system can include at least one of: a nuclear spin system, an electron spin system, a trapped-ion system, a neutral-atom system, or a superconducting circuit. In some embodiments, the one or more noise sources include at least two of: flux noise, charge noise, and photon shot noise. 
     According to one aspect of the present disclosure, a system may be provided for identifying and discriminating contributions from one or more noise sources using the multi-level structure of a quantum system with three or more levels. The system can include a signal generator and a noise identification processor. The signal generator may be configured to: prepare the quantum system in a predetermined state and apply one or more control signals to the quantum system. The noise identification processor may be configured to: measure values of one or more observables of the quantum system that quantify the quantum system&#39;s response to the noise sources and the one or more applied control signals; extract noise spectra information associated with the noise sources from the measured values; and identify contributions from the one or more noise sources based on the noise spectra information. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various objectives, features, and advantages of the disclosed subject matter can be more fully appreciated with reference to the following detailed description of the disclosed subject matter when considered in connection with the following drawings, in which like reference numerals identify like elements. 
         FIG. 1  is a block diagram of an illustrative noise identification system, which uses a multi-level quantum system as a noise sensor, according to disclosed embodiments. 
         FIG. 2A  is a diagram illustrating level transitions in a multi-level quantum system, according to disclosed embodiments. 
         FIGS. 2B and 2C  are graphical diagrams showing examples of spin-locking sequences that can be used to measure the relaxation of spin polarization in a multi-level quantum system, according to disclosed embodiments. 
         FIG. 3  is a circuit diagram of an illustrative superconducting circuit that may form part of a multi-level quantum system, according to disclosed embodiments. 
         FIG. 4  is a flow diagram of a process for identifying noise sources in multi-level quantum systems, according to disclosed embodiments. 
     
    
    
     The drawings are not necessarily to scale, or inclusive of all elements of a system, emphasis instead generally being placed upon illustrating the concepts, structures, and techniques sought to be protected herein. 
     DETAILED DESCRIPTION 
     Disclosed herein is a spin-locking-based quantum noise spectroscopy (QNS) protocol that exploits the multi-level energy structure of a superconducting qubit, along with various embodiments of systems and methods that utilize the protocol. Disclosed embodiments can extend the spectral range of weakly anharmonic qubit spectrometers beyond the present limitations set by their lack of strong anharmonicity (e.g., it expands the spectral range of a qubit spectrometer beyond the two-level approximation). In addition, using additional information gained from probing the higher-excited levels, disclosed embodiments can identify and distinguish contributions from different underlying noise mechanisms. That is, embodiments of the present disclosure allow for extracting the noise contributions from various noise sources, since the matrix elements for higher qubit transitions typically scale differently with level number depending on the origin of the noise. 
       FIG. 1  shows a noise identification system  100  in which noise contributions from various noise sources can be identified using multi-level noise spectroscopy, according to disclosed embodiments. The illustrative system  100  can include a signal generator  102 , a multi-level quantum system  104 , and a noise identification processor  106 . 
     Signal generator  102  can generate control signals  120  for the multi-level quantum system  104 . Control signals  120  can be, for example, pulsed sequences or continuous drive signals. Examples of control signal waveforms are shown and described in the context of  FIGS. 2B and 2C . Signal generator  102  can include a collection of hardware and/or software components configured to perform and execute the processes, steps, or other functionality described in conjunction therewith. 
     Multi-level quantum system  104  may include, for example, a nuclear spin system, an electron spin system, or superconducting circuit operating as a quantum computer. Multi-level quantum system  104  can include one or more weakly anharmonic qubits (and, in some embodiments, at least three qubits). Examples of weekly anharmonic qubits include the transmon (or “Xmon”), the gatemon, and the capacitively shunted flux qubit. In some embodiments, one or more of the qubits (e.g., a flux-tunable transmon qubit) may be operated as a multi-level noise sensor. Non-limiting examples of superconducting circuitry that can be provided within multi-level quantum system  104  is shown and described in the context of  FIGS. 3A and 3B . 
     Multi-level quantum system  104  can interact with one or more noise sources  130   a ,  130   b , . . .  130   n  ( 130  generally), which can result in decoherence in the multi-level system  104 . Noise sources  130  can include, for example, one or more of flux noise, charge noise, and photon shot noise. Noise sources  130  can include environmental noise. The types of noise that are present in a given environment may depend on the hardware makeup of the multi-level system. Likewise, the distinguishable noise sources may vary depending upon the multi-level system hardware. In general, the present disclosure can be used to distinguish and identify any type of noise so long as it is coupled to level transitions of the multi-level system with different relative coupling strength. 
     Noise identification processor  106  may be coupled to quantum system  104  and configured to measure the response of multi-level system  104  to the control signals  120 . That is, processor  106  may receive readout signals  122  from multi-level system  104  and, based on the readout signals  122 , measure a set of observables associated with multi-level system  104 . Processor  106  can extract information about the noise spectra and sources  130  from the measured values of the observables by reconstructing noise spectra at each level of the multi-level system  104 . Knowledge of the noise spectra enables identifying dominant noise sources affecting the multi-level system. Noise identification processor  106  can include a collection of hardware and/or software components configured to perform and execute the processes, steps, or other functionality described in conjunction therewith. 
     In some embodiments, multi-level system  104  (which serves as a noise probe) may be initialized (or “prepared”) in a known state. The multi-level system  104  may be allowed to evolve under conditions imposed by both environmental noise  130  and control signals  120  (e.g., pulse sequences or continuous drive signals). A detailed description of these steps, according to some embodiments, is provided below in the context of  FIGS. 2B and 2C . These steps can be taken prior to measuring a set of observables of the qubits that quantifies their response to noise sources  130  and the applied control  120 . 
     Having identified noise contribution, processor  106  can provide, as output  124 , information regarding the identified noise source(s). The noise spectra and identification information  124  can be used to improve coherence and gate performance within the multi-level quantum system  104 . In particular, information about noise sources  130  can be provided as an output  124  and used in an offline manner to improve the design and engineering of the multi-level system  105 . For example, the extracted noise spectra and noise source information can be used to increase a shunt capacitance of a superconducting qubit to make it insensitive to charge noise. As another example, extracted noise spectra and noise source information can be used to add more microwave filters along the control lines to reduce photon shot noise. These and other hardware modifications can result in a qubit system with better coherence. 
     Described next are multi-level quantum systems (or “sensors”) and a spin-locking-type QNS protocol that can used to identify noise sources in such systems/sensors. Embodiments of the QNS protocol described herein can be implemented within, for example, noise identification system  100  of  FIG. 1 . More particularly, portions of the protocol may be implemented within signal generator  102  and/or noise identification processor  106  of  FIG. 1 . 
     Referring to  FIG. 2A , consider an externally-driven M-level quantum system  200  (M&gt;2), which serves as the quantum noise sensor that evolves under the influence of its noisy environment (bath). That is, system  200  has levels  202   a ,  202   b ,  202   c ,  202   d , etc. For simplicity, only pure dephasing (σ z -type) noise may be considered in the following discussion. In the interaction picture with respect to the bath Hamiltonian H B , the joint sensor-environment system can be described by the Hamiltonian: 
                       H   ⁡     (   t   )       =     ℏ   ⁢       ∑     j   =   1       M   -   1       ⁢     [         (       ω   s     (   j   )       +       B     (   j   )       ⁡     (   t   )         )     ⁢        j   〉     ⁢     〈   j          +       ξ   ⁡     (   t   )       ⁢       λ     (       j   -   1     ,   j     )       ⁡     (       σ   +     (       j   -   1     ,   j     )       +     σ   -     (       j   -   1     ,   j     )         )           ]           ,           (   1   )               
where |j   j| is the projector for the j-th level of the multi-level sensor. The sensor eigen energies are ℏω s   (j)  with the ground state energy set to zero, and the B (j) (t) correspond to the time-dependent noise operators that longitudinally couple to the j-th level of the sensor and cause level j to fluctuate in energy. The raising and lowering operators of the sensor are denoted by σ +   (j−1,j) ≡|j   j−1| and σ −   (j−1,j) ≡|j−1   j|, respectively. The external driving field is denoted by ξ(t). In some embodiments, the multi-level sensor may be continuously driven with a signal ξ(t)=A drive  cos(ω drive t+ϕ), where A drive , ω drive , and ϕ correspond to the amplitude, the frequency and the phase of the driving field, respectively, and ϕ=0 may be assumed without loss of generality. The parameter λ (j−1,j)  represents the strength of the |j−1 −|j  transition relative to the |0 −|1  transition with λ (0,1) ≡1.
 
     When the drive frequency ω drive  is resonant with the |0 -|1  transition frequency ω s   (0,1)  of the sensor, the first two levels  202   a ,  202   b  form a pair of dressed states, |+ (0,1)    and |− (0,1)   . The level separation between dressed states is the Rabi frequency Ω (0,1)  as illustrated within box  204 , and it is determined predominantly (although not necessarily exactly, as describe below) by the effective driving strength λ (0,1) A drive ≡A drive . These dressed states form the usual spin-locking basis {|+ (0,1)   , |− (0,1)   } of a conventional, driven two-level sensor. 
     As illustrated in  FIG. 2A , the two-level spin-locking concept can be generalized to the case of a multi-level sensor  200 . By resonantly driving at the frequency ω s   (j−1,j) ≡ω s   (j) −ω s   (j−1)  of the transition between states |j−1  and |j , the system forms a pair of dressed states, |+ (j−1,j)    and |− (j−1,j)   , separated by a Rabi frequency Ω (j−1,j)  that is determined predominantly by an effective driving strength λ (j−1,j) A drive . For example, box  206  illustrates Rabi frequency Ω (1,2)  corresponding to the level separation between dressed states formed by the second and third levels  202   b ,  202   c . The effective two-level system formed by the basis {|+ (j−1,j)   , |− (j−1,j)   } acts as the j-th spectrometer and probes dephasing noise that leads to a fluctuation of the |j−1 −|j  transition at frequency Ω (j−1,j) . The case j=1 then corresponds back to the conventional two-level noise sensor. 
     In the present disclosure, the reference frame and two-dimensional subspace defined by the j-th spin-locking basis {|+ (j−1,j)   , |− (j−1,j)   } are referred to as the j-th “spin-locking frame” and the j-th “spin-locking subspace,” respectively. To move to the j-th spin-locking frame, unitary transformations can be applied, and the Hilbert space of the multi-level sensor can be truncated into the j-th spin-locking subspace. Then, the effective Hamiltonian describing the j-th noise spectrometer is: 
                           H   ~       S   ⁢   L       (       j   -   1     ,   j     )       ⁡     (   t   )       =           ℏ   2     ⁡     [       Ω     (       j   -   1     ,   j     )       +         B   ~     ∥     (       j   -   1     ,   j     )       ⁡     (   t   )         ]       ⁢       σ   ~     z     (       j   -   1     ,   j     )         +     ℏ   ⁢         B   ~     ⊥     (       j   -   1     ,   j     )       ⁡     (   t   )       ⁢     (         σ   ~     +     (       j   -   1     ,   j     )       +       σ   ~     -     (       j   -   1     ,   j     )         )           ,           (   2   )               
where {tilde over (σ)} z   (j−1,j) , {tilde over (σ)} +   (j−1,j) , and {tilde over (σ)} (j−1,j)  denote the Pauli Z operator, the raising operator, and the lowering operator of the j-th spin-locked spectrometer, respectively. The operators {tilde over (B)} ⊥   (j−1,j) (t) and {tilde over (B)} ∥   (j−1,j) (t) denote the noise operators that lead to longitudinal and transverse relaxation of the spectrometer within the j-th spin locking subspace, respectively. They are given as linear combinations of B (j) (t), arising from the level dressing across multiple levels as follows:
 
 {tilde over (B)}   ⊥   (j−1,j) ( t )=Σ k=1   M-1 α (j−1,j)   (k)   B   (k) ( t ),  (3)
 
 {tilde over (B)}   ∥   (j−1,j) ( t )=Σ k=1   M-1 β (j−1,j)   (k)   B   (k) ( t ),  (4)
 
where the so-called “noise participation ratio” α (j−1,j)   (k)  (β (j−1,j)   (k) ) is defined as a dimensionless factor that quantifies the fraction of the dephasing noise at the k-th level that is transduced (i.e., projected) to transverse (longitudinal) noise of the j-th pair of spin-locked states. The noise) participation ratios α (j−1,j)   (k)  and (β (j−1,j)   (k)  can be estimated by numerically solving for the dressed states in terms of the bare states |j . Note that the sign of the noise participation ratios can be either positive or negative, leading to the possibility for effective constructive and destructive interference between the noise operators B (k) (t).
 
     There are two noteworthy distinctions between a manifestly two-level system and a multi-level system. First, although the splitting energy ℏΩ (j−1,j)  between the j-th pair of dressed states (the j-th spin-locked states) is predominantly determined by the effective driving energy ℏ(λ (j−1,j)  A drive ), they are not universally equivalent. For an ideal two-level system within the rotating wave approximation, the Rabi frequency is indeed proportional to the effective driving field via the standard Rabi formula. However, this is not generally the case in a multi-level setting due to additional level repulsion from adjacent dressed states. Rather, in the multi-level setting of relevance here, the distinction between Ω (j−1,j)  and λ (j−1,j)  A drive  must be taken into account to yield an accurate estimation of the noise spectrum. 
     Second, as a consequence of the multi-level dressing, more than two noise operators B (k) (t) generally contribute to the longitudinal relaxation within a given pair of spin-locked states. In the limit λ (j−1,j)  A drive  is small compared to the sensor anharmonicities, the Eqs. (3)-(4) reduce to
 
 {tilde over (B)}   ⊥   (j−1,j) ( t )≈½[ B   (j−1) ( t )− B   (j) ( t )], {tilde over (B)}   ∥   (j−1,j) ( t )≈0,  (5)
 
which conform to the standard spin-locking noise spectroscopy protocol for a two-level sensor. However, as the effective drive strength λ (j−1,j)  A drive  increases, the contribution of peripheral bare states—other than |j−1  and |j —to the formation of the spin-locked states |+ (j−1,j)    and |− (j−1,j)    increases. As a result, in the large λ (j−1,j) Ω drive  limit, the multilevel dressing transduces the frequency fluctuations of more than two levels to the longitudinal relaxation within the j-th spin locking frame. Also, this multi-level effect contributes to the emergence of non-zero transverse relaxation B ⊥   (j−1,j) )(t), terms which would otherwise be absent within a two-level approximation.
 
     Turning to  FIGS. 2B and 2C , a multi-level noise spectroscopy protocol can include measuring the energy decay rate Γ 1ρ   (j−1,j)  (i.e., longitudinal relaxation rate) and the polarization  {tilde over (σ)} z   (j−1,j) (τ)  in the j-th spin-locking frame, and then using these quantities to extract the spectral density {tilde over (S)} ⊥   (j−1,j)  of the noise transverse to the spin-locking quantization axis. This in turn can be related to the longitudinal spectral density S ∥   (j−1,j)  that cases dephasing (i.e., transverse relaxation) in the original, undriven reference frame (the qubit “lab frame”). 
     The protocol can include preparing the multi-level sensor in the j-th spin-locked state |+ (j−1,j)    by applying a sequence of resonant π pulses [π (0,1) , π (1,2) , . . . π (j−2,j−1) ], which act to sequentially excite the sensor from the ground state |0  to state |j−1 . Then, a π (j−1,j) /2 pulse can be applied along the y-axis of the Bloch sphere, where the north and south poles now correspond to |j−1  and |j , respectively. The pulse acts to rotate the Bloch vector from the south pole to the x-axis, thereby placing the multi-level sensor in the j-th spin-locked state |+ (j−1,j)   =(|j−1)+|j )/√{square root over (2)}. Subsequently, a spin-locking drive with amplitude A drive  can be applied along the x-axis (collinear with the Bloch vector) at a frequency resonant with the |j−1 −|j  transition and for a duration τ. For example, referring to  FIGS. 2B and 2C , a π (0,1) /2 pulse and a π (1/2) /2 pulse can be used to measure the relaxation of spin polarization |+ (0,1)    and |+ (1,2)    as a function of the spin-locking duration τ, respectively. Then, the spin-locking drive with amplitude A drive  can be applied along the x-axis at frequencies ω s   (0,1)  and ω s   (1,2)  resonant with the |0 −|1  and |1 −|2  transitions, respectively. 
     By adiabatically turning on and off the drive, the state of the sensor can be kept within the j-th spin-locking subspace. Once the drive is off, a second π (j−1,j) /2 pulse is applied along the y-axis in order to map the spin-locking basis {|+ (j−1,j)   , |− (j−1,j)   } onto the measurement basis {|j , |j−1 }, and the qubit is then read out. This procedure can be repeated N times to obtain estimates for the probability of being in states {|j  and |j−1 }, which represent the probability of being in states {|+ (j−1,j)    and |− (j−1,j)   }, respectively. 
     The above protocol can be repeated as a function of τ in order to measure the longitudinal spin-relaxation decay-function of the j-th spin-locked spectrometer. For each τ, a normalized polarization of the spectrometer can be defined, 
                       〈         σ   ~     z     (       j   -   1     ,   j     )       ⁡     (   τ   )       〉     ≡           ρ     (       j   -   1     ,     j   -   1       )       ⁡     (   τ   )       -       ρ     (     j   ,   j     )       ⁡     (   τ   )               ρ     (       j   -   1     ,     j   -   1       )       ⁡     (   τ   )       +       ρ     (     j   ,   j     )       ⁡     (   τ   )             ,           (   6   )               
where ρ (j,j) (τ) denotes the population (the probability) of the j-th level. From the τ-dependence of  {tilde over (σ)} z   (j−1,j) (τ) , both the relaxation rate Γ 1ρ   (j−1,j)  of the spin polarization and the equilibrium polarization {tilde over (σ)} z   (j−1,j) (τ)| τ→∞  can be extracted. The values Γ 1ρ   (j−1,j)  and {tilde over (σ)} z   (j−1,j) (τ)| τ→∞  extracted from an experiment performed at a particular Rabi frequency Ω (j−1,j)  are related to the transverse noise PSD {tilde over (S)} ⊥   (j−1,j) (ω) at angular frequency ω=Ω (j−1,j)  as follows:
 
     
       
         
           
             
               
                 
                   
                     
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     Here, the transverse noise spectrum {tilde over (S)} ⊥   (j−1,j) (ω) is the Fourier transform of the two-time correlation function of the transverse noise operators acting on the spectrometer:
 
 {tilde over (S)}   ⊥   (j−1,j) (ω)=∫ −∞   ∞   dτe   −iωτ     {tilde over (B)}   ⊥   (j−1,j) (τ) {tilde over (B)}   ⊥   (j−1,j) (0) .  (9)
 
     In some embodiments, when performing noise spectroscopy measurements, the spin relaxation for the 1&#39;st and 2&#39;nd spin-locked noise spectrometers can be recorded. Then, the traces may be fit to an exponential decay, allowing for extraction of Γ 1ρ   (j−1,j)  and  {tilde over (σ)} z   (j−1,j) (t) | t→∞ . This may be repeated for various drive amplitude A drive  in order to reconstruct {tilde over (S)} ⊥   (j−1,j) (ω). 
     Turning to  FIG. 3 , a superconducting circuit  300  may form part of a multi-level quantum system, such as multi-level system  104  of  FIG. 1 , according to some embodiments. The illustrative circuit  300  includes a plurality of qubits  302   a ,  302   b ,  302   c  ( 302  generally). While the example of  FIG. 3  includes three qubits-first qubit  302   a , second qubit  302   b , and third qubit  302   c -greater numbers of qubits can be provided in other embodiments. In some embodiments, qubits  302  may be provided as transmon qubits. 
     One of the qubits  302  may be used to measure environmental noise, whereas the other qubits may be used for quantum computing or other purpose. In the example shown, third qubit  302   c  may include a flux-tunable transmon sensor  304  to measure flux noise and photon shot noise. For experimental purposes, circuit  300  can include independent channels  306  and  308  that can be used to apply flux noise and photon shot noise, respectively, to the superconducting circuit  300 . That is, known, engineered flux noise and photon-shot noise can be applied to the third qubit  302   a  via channels  306 ,  308 . In practice, such noise sources may be environmental, as shown in described above in the context of  FIG. 1 . 
     The noise-measuring transmon  304  can be controlled via a capacitively coupled control line  310 . The control line  310  may be coupled to receive control signals from a signal generator, such as signal generator  102  of  FIG. 1 . The additional qubits  302   a  and  302   b  may be strongly frequency-detuned from the transmon sensor  304 . 
     The circuit  300  can include a readout line  312  inductively coupled to channel  308 , also referred to as readout resonator  308 . The readout resonator  308  is capacitively coupled to the qubits  302  such that its resonant frequency is shifted depending on the quantum state of the qubits (this frequency shift is sometimes referred to as “dispersive shift”). A path comprising readout line  312  and readout resonator  308  enables read out of the quantum states of the qubits  302 . The readout line  312  can be coupled to a noise identification processor, such as processor  106  of  FIG. 1 . That is, readout line  312  can provide readout signals used by the noise identification processor to measure a set of observables associated with superconducting circuit  300  and, in turn, to identify dominant noise sources. Together, the control line  310  and readout line  312 , and readout resonator  308  can be used to perform noise spectroscopy protocol and measure the results using embodiments of the multi-level QNS protocol disclosed herein. 
     Control line  310  may be omitted in some embodiments, and transmon  304  may be controlled via a path including readout line  312  and readout resonator  308 , or via flux line  306 . 
     In an experimental setting, known, engineered photon-shot noise may also be applied to readout line  312 , which is inductively coupled to the readout resonator  308  as shown in  FIG. 3 . 
     In some embodiments, the disclosed multi-level QNS protocol can be used for accurate spectral reconstruction of engineered flux noise over a frequency range 50 MHz to 300 MHz, overcoming the spectral limitations imposed by a sensor&#39;s relatively weak anharmonicity (e.g., of approximately 200 MHz). Furthermore, by measuring the power spectra of dephasing noise acting on both the |0 -|1  and |1 -|2  transitions, noise contributions from both flux noise and photon shot noise can be extracted and uniquely identified, an attribution that is not possible within solely a two-level approximation. 
       FIG. 4  is a flow diagram of a process  400  for identifying noise contributions from various noise sources in a multi-level quantum system, according to disclosed embodiments. The illustrative process (or “protocol”) can be implemented and/or used within noise identification system  100  of  FIG. 1 . 
     At block  402 , the multi-level system, which serves as a noise probe, may be prepared in known state using, for example, techniques described above in the context of  FIGS. 2B and 2C . 
     At block  404 , one or more controls signals may be applied the multi-level system. The control signals may be generated by a signal generator, such as signal generator  102  of  FIG. 1 . The probe can be allowed to evolve under both environmental noise and the control signals. Here, the control signals can be either pulse sequences or continuous drive signals. Examples of control signal waveforms that may be used are shown and described in the context of  FIGS. 2B and 2C . 
     At block  406 , the response of multi-level system may be determined by measuring a set of observables of the qubits within the multi-level system. The measured observable values may quantify the qubits&#39; response to the environmental noise and the applied control. 
     At block  408 , information about the noise spectra and sources may be extracted from the measured values of the observables. In more detail, noise spectra at each level of the multi-level system may be reconstructed from the measured values. 
     At block  410 , one or more dominant noise sources may be identified using the extracted noise spectra information. Knowledge of the noise spectra and the dominant noise sources can then be used to improve coherence and gate performance within the multi-level quantum system using one of or more of the improvement techniques described above in the context of  FIG. 1 . 
     The subject matter described herein can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structural means disclosed in this specification and structural equivalents thereof, or in combinations of them. The subject matter described herein can be implemented as one or more computer program products, such as one or more computer programs tangibly embodied in an information carrier (e.g., in a machine-readable storage device), or embodied in a propagated signal, for execution by, or to control the operation of, data processing apparatus (e.g., a programmable processor, a computer, or multiple computers). A computer program (also known as a program, software, software application, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or another unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file. A program can be stored in a portion of a file that holds other programs or data, in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers at one site or distributed across multiple sites and interconnected by a communication network. 
     The processes and logic flows described in this specification, including the method steps of the subject matter described herein, can be performed by one or more programmable processors executing one or more computer programs to perform functions of the subject matter described herein by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus of the subject matter described herein can be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). 
     Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processor of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, or optical disks. Information carriers suitable for embodying computer program instructions and data include all forms of nonvolatile memory, including by ways of example semiconductor memory devices, such as EPROM, EEPROM, flash memory device, or magnetic disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. 
     It is to be understood that the disclosed subject matter is not limited in its application to the details of construction and to the arrangements of the components set forth in the following description or illustrated in the drawings. The disclosed subject matter is capable of other embodiments and of being practiced and carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting. As such, those skilled in the art will appreciate that the conception, upon which this disclosure is based, may readily be utilized as a basis for the designing of other structures, methods, and systems for carrying out the several purposes of the disclosed subject matter. Therefore, the claims should be regarded as including such equivalent constructions insofar as they do not depart from the spirit and scope of the disclosed subject matter. 
     Although the disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, it is understood that the present disclosure has been made only by way of example, and that numerous changes in the details of implementation of the disclosed subject matter may be made without departing from the spirit and scope of the disclosed subject matter.