Abstract
A logical qubit is a twodimensional subspace of a higher dimensional system, chosen such that it is possible to detect and correct the occurrence of certain errors. Manipulation of the encoded information generally requires arbitrary and precise control over the entire system. Whether based on multiple physical qubits or larger dimensional modes such as oscillators, the individual elements in realistic devices will always have residual interactions, which must be accounted for when designing logical operations. Here we demonstrate a holistic control strategy which exploits accurate knowledge of the Hamiltonian to manipulate a coupled oscillatortransmon system. We use this approach to realize highfidelity (98.5%, inferred), decoherencelimited operations on a logical qubit encoded in a superconducting cavity resonator using fourcomponent cat states. Our results show the power of applying numerical techniques to control linear oscillators and pave the way for utilizing their large Hilbert space as a resource in quantum information processing.
Introduction
Quantum error correction aims at the creation of logical qubits^{1, 2} whose information storage and processing capabilities exceed those of its constituent parts. Significant progress has been made toward quantum state preservation by repeated error detection using stabilizer measurements in trapped ions^{3, 4}, nitrogen vacancy centers^{5}, and superconducting circuits^{6,7,8}. In order to go beyond storage and to manipulate the encoded information, one must perform operations on the whole system in such a way that it results in the desired transformation within the twodimensional subspace defining the logical qubit. Any encoding scheme will consist of multiple interacting components where the system dynamics are not naturally confined within the logical subspace. Therefore, implementing operations requires carefully tailored controls which address each component of the system and manage their mutual interactions. Recent efforts have achieved this level of control and have demonstrated operations on a five qubit code in nuclear spin ensembles^{9} and a seven qubit code in trapped ions^{4}.
An alternative to logical qubit implementations based on multiple two level systems is to encode quantum information in continuous variable systems or oscillators, for which there are several schemes^{10, 11}. In particular so called “cat states”, which are superpositions of coherent states, can be used as the logical states of an encoded qubit^{12}. They are attractive because coherent states are eigenstates of the photon annihilation operator (\(\hat a\)) and therefore singlephoton loss induces simple, tractable errors^{13}.
Replacing several two level systems by an oscillator drastically reduces the hardware cost and complexity by requiring fewer components to fabricate and manipulate. However, introducing higher dimensional modes raises the issue of how to realize complete control over the system. Driving an isolated harmonic oscillator results in a displacement operation, which can only produce coherent states from the vacuum. Any oscillatorbased logical qubit scheme will require a richer class of operations, which one can access via coupling to a nonlinear system.
In the case of a frequencytunable qubit coupled to an oscillator with the JaynesCummings (JC) interaction (\({H_{{\rm{JC}}}} = {\sigma _ + }\hat a + {\sigma _  }{\hat a^\dag }\)), it has been demonstrated that it is possible to prepare arbitrary states in the oscillator^{14, 15}.
In the far offresonant case, where the JC interaction reduces to the dispersive Hamiltonian (\({H_{\rm{d}}}/\hbar = \chi {a^\dag }a\left e \right\rangle \left\langle e \right\)), a small set of operations acting on a timescale of 2π/χ is in principle sufficient for universal control^{16, 17} and has been used for nontrivial operations^{18, 19}. Generally, however, any approach decomposing an arbitrary operation into a sequence of elementary gates generates only a small subset of physically allowed control fields. It, therefore, suffers from two issues limiting the achievable fidelity. First, the constructed sequences may require an unacceptably large number of gates, limiting operations which are feasible in the presence of decoherence. Second, the idealized model used by a constructive approach typically fails to account for the existence of higher order Hamiltonian terms such as the Kerr nonlinearity \({H_{{\rm{Kerr}}}}/\hbar = \frac{K}{2}{\left( {{{\hat a}^\dag }} \right)^2}{\hat a^2}\) and spurious residual couplings in multiqubit systems.
In this work, we address these problems by considering a full model of the time dependent Hamiltonian in the presence of arbitrary control fields. Nuclear magnetic resonance experiments have shown that, if the available controls are universal, numerical optimization procedures can reliably solve the inversion problem of finding control fields to implement an intended operation. These optimal control algorithms, in particular the Gradient Ascent Pulse Engineering (GRAPE) method^{20, 21}, have been successfully employed in a variety of other fields^{22, 23}. Since GRAPE crucially depends on the model of the system, its successful application is powerful evidence that the Hamiltonian used accurately captures the system dynamics over a broad range of driving conditions.
Results
System and control protocol demonstration
The physical system used in our experiments is schematically depicted in Fig. 1a. The seamless aluminum λ/4 coaxstub cavity resonator^{24} with a fundamental frequency 4452.6 MHz has an energy relaxation time of 2.7 ms. A singlejunction transmon with transition frequency 5664.0 MHz and an harmonicity of 236 MHz is dispersively coupled to the oscillator, resulting in an interaction term \(\chi {\hat a^\dag }\hat a\left e \right\rangle \left\langle e \right\), with χ/2π = −2.2 MHz. Crucially, additional higher order terms are determined accurately through a separate set of calibration experiments (Supplementary Note 1, Supplementary Fig. 1, Supplementary Table 1). Control of the system is implemented through full inphase/quadrature modulated microwave fields centered on the transmon (oscillator) frequency and sent to the pin coupling to the transmon (oscillator) mode (setup schematic in Supplementary Fig. 2). In the rotating wave approximation, this results in the drive Hamiltonian H _{c}/ħ = ϵ_{C} a + ϵ_{T} σ _{_} + h.c. Modulation using an arbitrary waveform generator allows the coefficients ϵ_{C} and ϵ_{T} to be arbitrary complexvalued functions of time.
As an example application of GRAPE to our system (Supplementary Note 2), we find ϵ_{C}(t) and ϵ_{T}(t) such that, starting from the vacuum (Supplementary Fig. 3, Supplementary Note 3), after 500 ns of driven evolution the system ends up in the state \(\left {g,6} \right\rangle \), as shown in Fig. 1b, c. This highly nontrivial operation, effectively realizing a \(\left 6 \right\rangle \left\langle 0 \right\) coupling term on the oscillator, is enabled by the dispersive Hamiltonian using only linear drives on the transmon and the oscillator.
Encoding a logical catqubit
Using our control strategy, we can target the creation and manipulation of a logical qubit encoded in an evenparity fourcomponent cat subspace. Omitting normalization, the code states in this subspace can be written as
where we use \(\alpha = \sqrt 3 \). These code words are both of even photon number parity, and are distinguished by their photon number modulo 4:
Single photon loss, the dominant error channel for the system, transforms both code words to states of odd photon number parity. The encoded information, however, is preserved by this process as long as one can keep track of the number of photons that have been lost. Since parity measurements can be performed efficiently and nondestructively^{25}, single photon loss is a correctable error^{13}.
Using GRAPE, we create a universal set of gates on our logical qubit, which includes X and Y rotations by π and π/2, as well as Hadamard and T gates. These pulses are each 1100 ns ≈ 2.4 × 2π/χ in length with a 2 ns time resolution, although 99% of the spectral content lies within a bandwidth of 33 MHz (27 MHz) for the transmon (oscillator) drive (Supplementary Fig. 4). Each operation is designed to begin and end with the transmon in the ground state. Additionally, we create encode (U_{enc}) and decode (U_{dec}) pulses to transfer a bit of quantum information between our transmon \(\{ \left {g,0} \right\rangle ,{\kern 1pt} \left {e,0} \right\rangle \} \) subspace, which we can easily prepare and measure, and our encoded subspace \(\{ \left {g, + {{\rm{Z}}_{\rm{L}}}} \right\rangle ,{\kern 1pt} \left {g,  {{\rm{Z}}_{\rm{L}}}} \right\rangle \} \) (Fig. 2a).
We characterize the encode operation by preparing all six cardinal points on the transmon Bloch sphere, applying the encode pulse and performing Wigner tomography on the oscillator (Fig. 2b–d). Maximum likelihood reconstruction of the density matrix associated with the measured Wigner functions indicates an average state fidelity of 0.96. This metric underestimates the fidelity of U_{enc} because it is affected by several sources of error not intrinsic to the encoding operation itself, including error in the parity mapping and measurement infidelity.
Gate characterization
Process tomography provides a full characterization of a quantum operation, but depends on preexisting trusted operations and measurements which are not available for our encoded subspace. However, an indirect characterization of a gate U_{X} on our logical qubit can be performed using the operation U_{dec}U_{X}U_{enc}, which maps the transmon subspace onto itself. This allows one to use the trusted state preparations and measurements on the transmon to perform tomography on the composite process (Fig. 3a). The reconstructed process matrices in Fig. 3b show qualitative agreement with the intended encoded qubit gates. The process fidelities we report are average gate fidelities \({\cal F}({{\cal E}_1},{{\cal E}_2}) \equiv {\int} {\rm{d}}\psi F({{\cal E}_1}(\psi ),{{\cal E}_2}(\psi ))\), where F is the usual quantum state fidelity F(ρ _{1},ρ _{2}) = Tr(ρ _{1} ρ _{2}). We can break the measured infidelity down into three parts: transmon preparation and measurement error, encodedecode error and gate error. Using the experimentally determined process fidelities both without any operation \({{\cal F}_{{\rm{PT}}}}({\rm{No}}\,{\rm{Op}}.) = 0.982\), as well as with the encode and decode pulses \({{\cal F}_{{\rm{PT}}}}({{\rm{U}}_{{\rm{dec}}}}\,{{\rm{U}}_{{\rm{enc}}}}) = 0.964\), we estimate an infidelity contribution of approximately 1.8% for each of the first two components. To account for these factors to first order, the infidelity of operations on the encoded qubit are reported relative to \({{\cal F}_{{\rm{PT}}}}({{\rm{U}}_{{\rm{dec}}}}{{\rm{U}}_{{\rm{enc}}}})\). We find an average infidelity of 0.75% over our set of nine gates (Table 1).
In order to establish the fidelity of this set of operations more accurately, we perform randomized benchmarking^{26} (RB) on our encoded qubit (Fig. 4a). Careful analysis is required to infer the actual gate fidelity, as leakage out of the logical space in the oscillator does not present itself directly in the binary measurement of the state of the transmon qubit. Simulations show that such an RB experiment on a logical qubit with a larger associated Hilbert space typically overestimates the fidelity by a factor 1.7 ± 0.1 (Supplementary Fig. 6, Supplementary Note 4). From the resulting data (Fig. 4c) we infer an average gate fidelity of 0.985. This measurement is so sensitive to the quality of the applied gates that it is the ideal metric to use to finetune several experimental parameters (Supplementary Fig. 7, Supplementary Fig. 8, Supplementary Note 5). The infidelity of each of the individual gates is isolated using interleaved randomized benchmarking^{27} (iRB), which alternates between a single fixed and a random gate (Fig. 4b). Comparing the fitted decay constants of the RB and iRB results allows us to extract the fidelity of the fixed gate. The results are summarized in Table 1, together with the gate fidelities based on process tomography (Fig. 3) and Lindblad master equation simulations accounting for finite T _{1} and T _{2} of the transmon and oscillator. We note that all gates are implemented with an approximately equal infidelity of 1.5%. The results obtained using process tomography and iRB are not consistent, leading us to conclude that the approximation of the infidelity as the differrence between \({{\cal F}_{{\rm{PT}}}}({{\rm{U}}_{{\rm{dec}}}}{{\rm{U}}_{\rm{X}}}{{\rm{U}}_{{\rm{enc}}}})\) and \({{\cal F}_{{\rm{PT}}}}({{\rm{U}}_{{\rm{dec}}}}{{\rm{U}}_{{\rm{enc}}}})\) leads to an underestimation of the underlying infidelity of U_{X}. While several sources of decoherence are accounted for in the master equation simulations, the dominant source of infidelity in the model is transmon dephasing (T _{2} ≈ 43 μs). The good agreement between simulations and experiment indicates that the infidelity is primarily caused by decoherence and that additional contributions associated with imperfections in the model Hamiltonian and the applied pulses are a significantly smaller effect.
Discussion
A logical quantum bit consists of a quantum system with multiple degrees of freedom that can be used to correct for a finite set of errors, at the expense of being more complicated to control. Before one can realize the ultimate goal of robust and high fidelity operations that surpass the performance of the physical qubits, there are several challenging steps which must be demonstrated. First one must devise a code that can detect and correct for the dominant errors, second is to implement the code by demonstrating encoding and decoding operations, third is to show the ability to detect and correct the errors, and fourth is to manipulate the information in the encoded system by performing logical operations. Finally, one must eventually combine all of these components to improve the systems lifetime and the fidelity of operations.
In this work, we show an important step along this path, namely the first manipulations of a logical qubit encoded in catstates. At this level one should generally expect the overall error rate to actually increase relative to the physical components. This is due to the additional overhead of implementing a logical qubit, which originates from the redundant encoding required to detect and correct errors. For instance, the seven qubits in the Steane code^{28} result in an effective error rate, which is larger by a factor 7. Our scheme, the catcode using several levels in an oscillator, increases the error rate by a factor \(\bar n \approx 3\) in the best case scenario.
The increased system complexity of a logical qubit poses additional challenges and sources of errors when performing logical operations. First of all, implementing any of the operations is nontrivial as the logical basis states are typically not easily constructed using the available controls. In fact, this is a key property of the encoding as it prevents the information to decohere due to interaction with the environment. When starting from the system’s ground state, for example, a sequence of several single and twoqubit gates is required to produce a logical basis state in the Steane code, or a complex pulse in our scheme. Additionally, when integrating many physical systems it remains an open experimental question whether controlfield crosstalk and interqubit interactions, which introduce coherent and correlated errors, can be engineered sufficiently small. Therefore, demonstrating accurate manipulation of a logical qubit is an important and necessary first step toward errorcorrected quantum computation. Such experiments lead to a better understanding of experimental nonidealities and other sources of errors.
By mapping a bit of quantum information from the transmon onto catstates in the oscillator, we have transferred the information onto a system with a coherence time which is more than an order of magnitude larger (Supplementary Fig. 9). The overhead of encoding/decoding is approximately 3%, and, therefore, it is beneficial when storing a state for more than around 3% of the transmon coherence time, or 1.3 μs. The dispersive coupling to the transmon still allows the information encoded in the logical qubit to be manipulated directly, albeit with a fidelity smaller than that of transmon operations. There are a number of protocols where this tradeoff between operation fidelity and lifetime is desirable, such as entanglement distillation^{29} and quantum repeaters^{30}. Photonloss error correction can enhance the lifetime of the logical catcode qubit even further^{13}, but would not enhance the operation fidelity. The reason is twofold: first of all, the system is not in a catstate during the gate operation (Supplementary Fig. 10), and second, photonloss is a much less significant source of errors than transmon dephasing. Although improving the transmon coherence time would directly result in higher fidelity gates, it is likely that this will remain the dominant source of errors.
However, simulations show that a large fraction of the errors that occur during an operation are detectable (Supplementary Table 2), and could, therefore, be mitigated using erasure correcting codes^{31}. This originates from the fact that the oscillatortransmon system takes a complicated trajectory through its Hilbert space during an operation. An error drastically alters this trajectory and, therefore, the final state at the end of the operation. For example, in approximately 50% of the cases an error occurs, the transmon will be left in the excited state. Additionally, it is unlikely that the oscillator will remain in the logical subspace after an error, and if this could be measured efficiently it would imply that, for the designed pulses, approximately 95% of the errors are detectable.
It might be possible to increase the fraction of detectable errors by optimizing control pulses in the presence of decoherence, an established technique^{32}, combined with an appropriate modification the cost function; we have not yet thoroughly explored this approach. A more fundamental open question is whether the GRAPE algorithm can be used to design pulses which implement operations faulttolerantly (e.g., with respect to transmon dephasing).
In conclusion, we have demonstrated a highfidelity implementation of a universal set of gates on a qubit encoded into an oscillator using the catcode. The low error rates for these operations are verified using both process tomography and randomized benchmarking, and the results are consistent with simulations which account for decoherence. We obtained these operations by numerically optimizing timedependent drives which make use of the wellcharacterized dispersive interaction between the far detuned oscillator and transmon modes. While in this Article we have focused on realizing and characterizing singlequbit operations on catencoded qubits, this control technique is not restricted to these goals, and is in principle capable of crafting arbitrary unitary operations on the transmonoscillator system. The high quality of these operations depends critically on an accurate characterization of the system Hamiltonian, and demonstrates the utility of numerical optimal control for realizing quantum information processing.
Data availability
Relevant data is available from R.W.H. upon request.
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Acknowledgements
We would like to thank Katrina Sliwa and Michael Hatridge for providing the parametric amplifier, Chris Axline, Jacob Blumoff, Kevin Chou, and Chen Wang for discussions regarding sample design, Stefan Krastanov, Chao Shen and Victor Albert for discussions on universal control and Steve Flammia and Robin BlumeKohout for advice about tomography. This research was supported by the U.S. Army Research Office (W911NF141011). P.R. was supported by the U.S. Air Force Office of Scientific Research (FA95501510015), L.J. by the Alfred P. Sloan Foundation and the Packard Foundation. Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering (YINQE), the Yale SEAS cleanroom, and the National Science Foundation (MRSECDMR1119826).
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R.W.H. and P.R. performed the experiment and data analysis under the supervision of R.J.S. P.R. developed the GRAPE implementation, which created the optimal control pulses. N.O. developed the Field Programmable Gate Array hardware, which controls the experiment. M.H.D. and L.J. provided theoretical support. R.W.H. and L.F. fabricated the transmon qubit. R.W.H., P.R., and R.J.S. wrote the manuscript with contributions from all authors.
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R.J.S., M.H.D., and L.F. are equity holders and consultants at Quantum Circuits, Inc. R.W.H., P.R., L.J., L.F., and R.J.S. are coinventors on a patent submission by Yale University related to this work. N.O. declares no competing financial interest.
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Heeres, R., Reinhold, P., Ofek, N. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat Commun 8, 94 (2017). https://doi.org/10.1038/s41467017000451
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