Patent Description:
This invention was made with government support under FA9550-<NUM>-<NUM>-<NUM> awarded by the United States Air Force Office of Scientific Research and under W911NF-<NUM>-<NUM>-<NUM> awarded by the United States Army Research Office. The government has certain rights in the invention.

Communication channels and data memory systems are often subject to sources of noise, which can introduce errors into the stored or transmitted data. Classical error detection and correction schema add redundancy to the transmitted or stored information to enable reliable delivery or recovery of digital data. Such classical schema can sample the transmitted or stored information to determine if the data has been corrupted and/or to correct the corrupted data.

However, quantum information processing uses quantum mechanical phenomena, such as energy quantization, superposition, and entanglement, to encode and process information in a way not utilized by conventional information processing. For example, it is known that certain computational problems may be solved more efficiently using quantum computation rather than conventional classical computation. Such quantum information processing cannot utilize classical error correction procedures because measuring the state of the quantum information system will collapse any superposition and/or entangled states, per the Heisenberg uncertainty principle. Scalable quantum computation will ultimately require quantum error correction schemes to provide fault-tolerant quantum information processing.

<NPL>) teaches techniques for quantum error correction based on quantum position operators.

Continuous variable bosonic quantum systems are ubiquitous in various quantum computing and communication architectures and provide unique advantages to, for example, quantum simulation of bosonic systems such as boson sampling and vibrational quantum dynamics of molecules. Continuous variable bosonic quantum information processing may be implemented by using harmonic oscillator modes in a photonic system (e.g., quantum optics systems) or a phononic system (e.g., trapped ion systems). However, such harmonic oscillator modes may experience noise caused by, for example, excitation loss and/or thermal noise errors, which may alter or corrupt information stored in the quantum information system. Such noise sources can be challenges for realizing large-scale and fault-tolerant continuous variable quantum information processing.

Quantum error correction (QEC) schema, which may detect and/or correct for introduced errors caused by environmental noise, are therefore important for implementing scalable and fault-tolerant quantum information processing. In many bosonic QEC schemes, a finite-dimensional quantum system (e.g., a qubit) is encoded into an oscillator or into many oscillators. However, in such qubit-into-an-oscillator encoding schemes, the infinite-dimensional bosonic nature of the physical oscillator modes is lost at the logical level because the error-corrected logical system is described by discrete variables such as Pauli operators. Therefore, the error-corrected logical discrete variable system is not itself tailored to continuous-variable quantum information processing tasks.

The inventors have recognized and appreciated that if an infinite-dimensional oscillator mode is encoded into many noisy oscillator modes (e.g., oscillator-into-oscillators encoding), such an error-corrected oscillator mode will still be tailored to various continuous-variable quantum information processing tasks. In general, oscillator-into-oscillators bosonic QEC is more challenging than qubit-into-oscillators bosonic QEC because in the former, an infinite-dimensional bosonic Hilbert space is being protected against relevant errors, whereas in the latter only a finite-dimensional Hilbert space embedded in infinite-dimensional bosonic modes is being protected against relevant errors. For this reason, some qubit-into-oscillators QEC codes exist which can correct experimentally-relevant Gaussian errors such as excitation losses, thermal noise, and additive Gaussian noise errors. However, some previously-proposed oscillator-into-oscillators QEC codes cannot correct Gaussian errors because they rely on Gaussian QEC schemes, and established no-go theorems state that Gaussian errors cannot be corrected using only Gaussian resources because of the Heisenberg uncertainty principle.

The Heisenberg uncertainty principle states that the position and momentum operators <MAT> and <MAT> cannot be measured simultaneously because they do not commute with each other (e.g., [q̂, p̂] = i ≠ <NUM>). However, the following displacement operators do commute with each other: <MAT> and therefore can be measured simultaneously. Note that measuring <MAT> and Ŝp = <MAT> is equivalent to measuring their exponents (or phase angles) <MAT> and <MAT> modulo <NUM>πi. Thus, the commutativity of Ŝq and Ŝp implies that the position and momentum operators can be measured simultaneously if they are measured modulo <MAT>.

The inventors have recognized and appreciated that these no-go theorems can be circumvented by using non-Gaussian resources to perform bosonic QEC. Examples of non-Gaussian resources include the single-photon Fock state and photon-number-resolving measurements, Kerr nonlinearities, cubic phase state and gate, selective number-dependent arbitrary phase (SNAP) gate, Schrödinger cat states, and Gottesman-Kitaev-Preskill (GKP) states. The inventors have further recognized and appreciated that GKP states may be particularly useful for performing oscillator-into-oscillators QEC, enabling fault-tolerant, continuous-variable bosonic systems.

The canonical GKP state (or the grid state) is defined as the unique (up to an overall phase) simultaneous eigenstate of the two commuting displacement operators Ŝq and Ŝp with unit eigenvalues. Explicitly, the canonical GKP state is given by: <MAT> and thus has no definite values for both the position and momentum operators modulo <MAT> (e.g., <MAT>).

The canonical GKP state has an infinite average photon number because it comprises superpositions of infinitely many ( <MAT>) infinitely squeezed states ( <MAT> or <MAT>). However, an approximate GKP state with a finite average photon number can be defined by applying a non-unitary operator exp [-Δn̂] to the canonical GKP state and then normalizing the output state: |GKPΔ〉 ∝ exp[-Δn̂]|GKP〉. <FIG> illustrates the Wigner function of the canonical GKP state with an average photon number n = <NUM>. Negative peaks (e.g., peak <NUM>) in the Wigner function in addition to positive peaks (e.g., peak <NUM>) indicate that the canonical GKP state is a non-Gaussian state.

Accordingly, the inventors have developed systems and methods for performing QEC using one or more GKP states. Some embodiments are directed to a quantum information device. The quantum information device comprises a plurality of quantum mechanical oscillators (e.g., microwave resonator cavities, trapped ion oscillators, or any suitable quantum oscillator). The plurality of quantum mechanical oscillators comprises a first data quantum mechanical oscillator configured to store information representing a first data state and a first ancilla quantum mechanical oscillator coupled to the first data quantum mechanical oscillator. The first ancilla quantum mechanical oscillator is configured to store information representing a first ancilla state, wherein the first ancilla state may be initialized as a first GKP state.

The quantum information device further comprises at least one electromagnetic radiation source coupled to the plurality of quantum mechanical oscillators. For example, the at least one electromagnetic radiation source may include a microwave source. The at least one electromagnetic radiation source is configured to entangle and disentangle the information stored in the first data quantum mechanical oscillator and the information stored in the first ancilla quantum mechanical oscillator by applying one or more drive waveforms to one or more of the quantum mechanical oscillators of the plurality of quantum mechanical oscillators.

At least one measurement device is coupled to the first ancilla quantum mechanical oscillator. The at least one measurement device is configured to measure a position operator and a momentum operator of the first ancilla quantum mechanical oscillator in order to detect error in the first data quantum mechanical oscillator. For example, the at least one measurement device may be at least one homodyne detector configured to perform homodyne detection to measure a position and a momentum operator of the first ancilla quantum mechanical oscillator.

It is also disclosed a method of performing QEC. The method comprises initializing a first GKP state in a first ancilla quantum mechanical oscillator. For example, the first GKP state may be initialized using a drive waveform (e.g., a microwave signal) or any other suitable technique. Then, information stored in a first data quantum mechanical oscillator is entangled with information stored in the first ancilla quantum mechanical oscillator by using an entangling operation (e.g., a SUM gate, a squeezing operation). The first data quantum mechanical oscillator and first ancilla quantum mechanical oscillator are then be disentangled by using a disentangling operation (e.g., an inverse SUM gate, an inverse squeezing operation). The position and momentum quadrature operators of the first ancilla quantum mechanical oscillator are measured using any suitable method, including, for example, homodyne detection. From the measured position and momentum quadrature operators, information indicative of noise in the information stored in the first data quantum mechanical oscillator is determined. This determined information indicative of noise is used to correct for noise in the information stored in the first data quantum mechanical oscillator, by applying a displacement operation (e.g., using a drive waveform) to the first data quantum mechanical oscillator.

Following below are more detailed descriptions of various concepts related to, and embodiments of quantum error correction techniques. It should be appreciated that various aspects described herein may be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only.

<FIG> illustrates an example of a bosonic system <NUM> configured for performing error correction, in accordance with embodiments described herein. Bosonic system <NUM> includes data oscillator <NUM> and ancilla oscillator <NUM> coupled to electromagnetic radiation source <NUM>. Data oscillator <NUM> and ancilla oscillator <NUM> may be any suitable bosonic quantum mechanical oscillator. For example, data oscillator <NUM> and ancilla oscillator <NUM> may be resonator cavities (e.g., microwave resonator cavities) or trapped ion oscillators. In some embodiments, data oscillator <NUM> and ancilla oscillator <NUM> may be coaxial resonator cavities formed from, for example, aluminum.

Electromagnetic radiation source <NUM> is configured to apply one or more electromagnetic signals to drive data oscillator <NUM> and/or ancilla oscillator <NUM>. Such electromagnetic signals may be, for example, microwave signals or laser-generated signals. The choice of electromagnetic signals used to drive data oscillator <NUM> and/or ancilla oscillator <NUM> may be determined based on the types of quantum mechanical oscillators used in bosonic system <NUM>, in some embodiments.

In some embodiments, the ancilla oscillator <NUM> may be put into an initial GKP state by an electromagnetic signal from electromagnetic radiation source <NUM>. The initial GKP state of the ancilla oscillator <NUM> is then be entangled with the state of the data oscillator <NUM>. Such entanglement is performed by one or more electromagnetic signals from electromagnetic radiation source <NUM>. For example, a SUM gate or a squeezing operation may be used to entangle the states of the data oscillator <NUM> and the ancilla oscillator <NUM>. A squeezing operation may be, for example, an operation configured to produce a squeezed state (e.g., wherein the squeezed state comprises a quantum uncertainty smaller than that of a coherent state).

After the entangled states of the data oscillator <NUM> and the ancilla oscillator <NUM> have encountered a source of noise such that both states include a correlated noise error (e.g., both states include the same error), the entangled states are disentangled and the error is detected and corrected. Disentangling of the states is performed by one or more electromagnetic signals from electromagnetic radiation source <NUM>. For example, an inverse SUM gate or an inverse squeezing operation may be used to disentangle the states of the data oscillator and the ancilla oscillator <NUM>.

The correlated noise error is detected by measuring position and momentum operators of the ancilla oscillator <NUM> so as to not disturb the state of the data oscillator <NUM>. The position and momentum operators of the ancilla oscillator <NUM>, in some embodiments, may be measured modulo <MAT>. In some embodiments, the position and momentum operators of the ancilla oscillator <NUM> may be measured, for example, using a homodyne detection measurement. Such homodyne detection measurements may be performed, in some embodiments, by a homodyne detector (e.g., an integrated photonic homodyne detector, or any suitable homodyne detector).

The measured position and momentum operators of the ancilla oscillator <NUM> is used to correct the error present in the state of the data oscillator <NUM>. Displacement operations are applied to the state of the data oscillator <NUM> by applying one or more electromagnetic signals from electromagnetic radiation source <NUM>.

In some embodiments, the state of the data oscillator <NUM> may be detected after performing a QEC procedure by readout resonator (not pictured) coupled to the data oscillator <NUM>. For example, in embodiments where the data oscillator <NUM> comprises a microwave resonator cavity, the readout resonator may include a stripline resonator that may be coupled to the data oscillator <NUM>.

<FIG> illustrates an additional example of a bosonic system <NUM> configured for performing error correction. Bosonic system <NUM> includes data oscillator <NUM>, first ancilla oscillator <NUM>, and second ancilla oscillator <NUM> coupled to electromagnetic radiation source <NUM>. Data oscillator <NUM>, first ancilla oscillator <NUM>, and second ancilla oscillator <NUM> may be any suitable bosonic quantum mechanical oscillator, including those discussed in connection with the example of <FIG>. It may be appreciated that the embodiment of <FIG> presents an example, and that additional embodiments may include any number of data and/or ancilla oscillators (e.g., greater than two data and/or ancilla oscillators, three data and/or ancilla oscillators, up to <NUM> data and/or ancilla oscillators, etc.).

Electromagnetic radiation source <NUM> is configured to apply one or more electromagnetic signals to drive data oscillator <NUM>, first ancilla oscillator <NUM>, and/or second ancilla oscillator <NUM>. Such electromagnetic signals may be, for example, microwave signals or laser-generated signals. The electromagnetic signals may be configured to, for example, initialize states within the oscillators, perform operations on the states within the oscillators, and/or entangle states of one or more of the oscillators of bosonic system <NUM>.

In some embodiments, the first ancilla oscillator <NUM> and the second ancilla oscillator <NUM> may be put into initial GKP states by electromagnetic signals from electromagnetic radiation source <NUM>. The initial GKP state of the first ancilla oscillator <NUM> is then entangled with the state of the data oscillator <NUM>, which is performed by one or more electromagnetic signals from electromagnetic radiation source <NUM>. For example, a SUM gate or a squeezing operation may be used to entangle the states of the data oscillator <NUM> and the first ancilla oscillator <NUM>.

After the entangled states of the data oscillator <NUM> and the first ancilla oscillator <NUM> have encountered a source of noise such that both states include a correlated noise error (e.g., both states include the same error), the entangled states are disentangled and the error may be detected and corrected. Disentangling of the states is performed by one or more electromagnetic signals from electromagnetic radiation source <NUM>. For example, an inverse SUM gate or an inverse squeezing operation may be used to disentangle the states of the data oscillator <NUM> and the first ancilla oscillator <NUM>.

In some embodiments, after disentangling the state of the first ancilla oscillator <NUM> from the state of the data oscillator <NUM>, the state of the first ancilla oscillator may be entangled with the state of the second ancilla oscillator <NUM>. Such entanglement may be performed by one or more electromagnetic signals from electromagnetic radiation source <NUM>. For example, a SUM gate may be used to entangle the states of the first ancilla oscillator <NUM> and the second ancilla oscillator <NUM>.

In some embodiments, the correlated noise error may be detected by measuring the position operator of the first ancilla oscillator <NUM> and the momentum operator of the second ancilla oscillator <NUM>. The position and momentum operators of the first ancilla oscillator <NUM> and the second ancilla oscillator <NUM>, in some embodiments, may be measured modulo <MAT>. In some embodiments, the position and momentum operators of the first ancilla oscillator <NUM> and the second ancilla oscillator <NUM> may be measured, for example, using a homodyne detection measurement. Such homodyne detection measurements may be performed, in some embodiments, by homodyne detectors <NUM> and <NUM>. Homodyne detectors <NUM> and <NUM> may be, for example, an integrated photonic homodyne detector, or any suitable homodyne detector.

In some embodiments, the measured position and momentum operators of the first ancilla oscillator <NUM> and the second ancilla oscillator <NUM> may be used to correct the error present in the state of the data oscillator <NUM>. For example, displacement operations may be applied to the state of the data oscillator <NUM>. In some embodiments, such displacement operations may be applied to the state of the data oscillator <NUM> by applying one or more electromagnetic signals from electromagnetic radiation source <NUM>.

<FIG> is a flowchart illustrating a process <NUM> for performing QEC. Process <NUM> may start at act <NUM>, in which a first GKP state may be initialized in a first ancilla quantum mechanical oscillator. For example, as described in connection with <FIG> and <FIG>, the first ancilla quantum mechanical oscillator may be any suitable bosonic quantum mechanical oscillator (e.g., a resonator cavity, a trapped ion oscillator, etc.). The first GKP state may be initialized in the first ancilla quantum mechanical oscillator using one or more electromagnetic signals (e.g., microwave signals, laser-generated signals).

In act <NUM>, the information (e.g., data) stored in a first data quantum mechanical oscillator is entangled with the information (e.g., the first GKP state) stored in the first ancilla quantum mechanical oscillator. In some embodiments, the entangling may be performed using a SUM gate or a squeezing operation implemented by, for example, one or more electromagnetic signals sent to the first data quantum mechanical oscillator and/or the first ancilla quantum mechanical oscillator.

After exposure to a source of noise, in act <NUM> the information stored in the first data quantum mechanical oscillator and the information stored in the first ancilla quantum mechanical oscillator are disentangled, which is performed using an inverse SUM gate or an inverse squeezing operation implemented by, for example, one or more electromagnetic signals sent to the first data quantum mechanical oscillator and/or the first ancilla quantum mechanical oscillator.

In act <NUM>, position and momentum quadrature operators of the first ancilla quantum mechanical oscillator are measured. The position and momentum quadrature operators of the first ancilla quantum mechanical oscillator may be measured, in some embodiments, using one or more homodyne detectors to perform homodyne detection. The position and momentum quadrature operators may be measured, for example, modulo <MAT> using homodyne detection.

In act <NUM>, the measured position and momentum quadrature operators of the first quantum mechanical oscillator are used to correct for an error in the information stored in the first data quantum mechanical oscillator. Displacement operations based on the measured position and momentum quadrature operators are applied to the first data quantum mechanical oscillator to correct for an error in the stored information. The displacement operations are applied by driving the first data quantum mechanical oscillator with one or more electromagnetic signals.

Turning back to <FIG>, which depicts the Wigner function of the canonical GKP state with an average photon number n = <NUM>, there have been many proposals for preparing an approximate GKP state in various experimental platforms. Notably an approximate GKP state has been realized in a trapped ion system and a variation of an approximate GKP state has been realized in a circuit quantum electrodynamics (QED) system. For the discussion herein in connection with <FIG>, the use of the ideal canonical GKP state is considered. The use of approximate GKP states will be discussed in connection with <FIG>.

The ability to measure the position and momentum operators modulo <MAT> allows for the preparation of the canonical GKP state. Additionally, the converse is also true. That is, the quadrature operators can be measured modulo <MAT> given GKP states and Gaussian operations as resources. As shown in <FIG>, the position and momentum operators of a data state |ψ〉 can be measured modulo <MAT> using a canonical GKP state, the SUM gate <NUM> or inverse SUM gate <NUM>, and a homodyne measurement <NUM> of the position operator and a homodyne measurement <NUM> of the momentum operator, respectively. In some embodiments, the SUM gate <NUM> is a Gaussian operation and is defined as SUMj→k ≡ exp[-iq̂jp̂k] which maps q̂k to q̂k + q̂j. The inverse SUM gate is defined as the inverse of the SUM gate, as defined herein. The canonical GKP state and the modulo simultaneous quadrature measurement are the non-Gaussian resources of the oscillator-into-oscillators encoding schemes, introduced below.

From these non-Gaussian resources, a non-Gaussian oscillator-into-oscillators code may be constructed. This code may be, in some embodiments, the two-mode GKP-repetition code, which may be used to correct additive Gaussian noise errors. <FIG> illustrates an example of a two-mode bosonic system <NUM> performing encoding ("entangling") of a data state, |ψ〉, and an ancilla state initialized in a GKP state, |GKP〉, using a two-mode GKP-repetition code. The data state and the ancilla state may be entangled using a SUM gate <NUM>. The two-mode bosonic system <NUM> may be subject to correlated Gaussian noise errors <NUM> after entangling of the two modes.

Quantum error-correcting codes function by hiding the quantum information from the environment by storing the logical state in a non-local entangled state of the physical components. In the case of the scheme presented in connection with <FIG>, a data oscillator state is entangled via a Gaussian operation with an ancillary oscillator mode, which is initially in the canonical GKP state, in a manner that prevents the environment from learning about the logically encoded state. Like qubit-into-an-oscillator GKP codes, this oscillator-into-oscillators code is specifically designed to protect against random displacement errors. Additionally, this scheme succeeds because of the assumption of access to non-Gaussian resources (e.g., GKP states, modular quadrature measurements) that are unavailable to the environment.

An arbitrary bosonic data state, |ψ 〉 = ∫ dqψ(q)|q̂<NUM> = q〉, may be encoded into two oscillator modes such that: <MAT> where |GKP〉 is the canonical GKP state in the second mode, or ancilla state. This encoding process is the two-mode GKP-repetition code described in connection with <FIG>. After entangling the data state and the ancilla state through the encoding process, the information of the data state may be stored in both the data state and the ancilla state. Further, the SUM gate, SUMj→k = exp[—iq̂jp̂k](j ≠ k), is a continuous variable analog of the CNOT gate, which, in the Heisenberg regime, transforms q̂k into q̂k + q̂j and p̂j into p̂j - p̂k while leaving all other quadrature operators unchanged. The quadrature operators are transformed by the encoding circuit into the following: <MAT> <MAT> <MAT> <MAT> Note that the SUM gate in the encoding circuit is analogous to the CNOT gate in the encoding circuit of the two-bit repetition code for qubit bit-flip errors. Additionally, the two-mode GKP-repetition code can be considered as a non-Gaussian modification of the two-mode Gaussian-repetition code.

The performance of the two-mode GKP-repetition code can be analyzed by assuming that the data and ancilla states undergo independent and identically distributed (iid) additive Gaussian noise errors (or Gaussian random displacement errors): <IMG>[σ] ⊗ <IMG>[σ]. Here, <IMG>[σ] is an additive Gaussian noise error acting on the kth mode which, in the Heisenberg regime, adds Gaussian random noise <MAT> and <MAT> to the position and momentum operators of the kth mode, e.g.,: <MAT>
where k ∈ (<NUM>,<NUM>). Also, <MAT> and <MAT> are independent Gaussian random variables with zero mean and variance σ<NUM>. That is, <MAT>. Such additive Gaussian noise errors may be generic in the sense that any excitation loss and/or thermal noise errors can be converted into an additive Gaussian noise error by applying a suitable quantum-limited amplification channel. For example, a pure excitation loss error with loss probability γ can be converted into an additive Gaussian noise error <IMG>[σ] with <MAT>.

<FIG> illustrates an example of the two-mode bosonic system <NUM> performing a decoding operation ("disentangling") using a two-mode GKP-repetition code. The data state and the ancilla state may be disentangled using an inverse SUM gate <NUM> (e.g., <MAT>). Upon the inverse SUM gate, the transformed quadrature operators are transformed back to the original quadrature operators but the added quadrature noise <MAT> and <MAT> may be reshaped, e.g., <MAT> <MAT>
for k ∈ {<NUM>, <NUM>}, where the reshaped quadrature noise <MAT> and <MAT> are given by: <MAT> <MAT> <MAT> <MAT> The position quadrature noise of the data mode, <MAT>, may be transferred to the position quadrature of the ancilla mode (see <MAT> in <MAT>).

Thereafter, momentum and position quadrature operators of the ancilla state may be measured modulo <MAT> by measurement devices <NUM> and <NUM> of <FIG>. In some embodiments, measurement devices <NUM> and <NUM> may be homodyne detectors and measurement of the momentum and position quadrature operators may be performed using a homodyne detection. By performing such measurements, both <MAT> and <MAT> are measured modulo <MAT>. Such measurements of <MAT> and <MAT> are equivalent to measurements of the reshaped ancilla quadrature noise <MAT> and <MAT> modulo <MAT> because the ancilla state was initially in the canonical GKP state such that <MAT> holds. The extracted information about <MAT> <MAT> and <MAT> may then be used to estimate the data position quadrature noise <MAT> and the ancilla momentum quadrature noise <MAT> such that the uncertainty of the data position quadrature noise is reduced while the ancilla momentum quadrature noise (transferred to the data momentum quadrature) does not degrade the momentum quadrature of the data mode.

From the outcomes of the measurements of <MAT> and <MAT> modulo <MAT>, the true values of <MAT> and <MAT> may be assumed to be the ones with the smallest length among the candidate values that are compatible with the modular measurement outcomes. That is, <MAT> where Rs(z) = z - n*(z)s and <MAT>. More concretely, Rs(z) equals a displaced sawtooth function with an amplitude and period s that is given by Rs(z) = z if z ∈ [-s/<NUM>, s/<NUM>]. Then, based on these estimates, it can be further estimated that the position quadrature noise of the data mode <MAT> and the momentum quadrature noise of the ancilla mode <MAT> are given by: <MAT> The former estimate may be based on a maximum likelihood estimation, in some embodiments, and as described in Appendix A herein.

The noise error in the data state is corrected based on the estimates of <MAT> and <MAT>. This correction may be performed, for example, by applying counter displacement operators, <MAT> and <MAT>, to the data state. The counter displacement operators may be momentum and position displacement operators <NUM> and <NUM>. In some embodiments, the displacement operators <NUM> and/or <NUM> may be applied to the data state using an electromagnetic signal (e.g., a microwave signal, a laser-generated photonic signal).

After applying the counter displacement operations to the data state, the logical position and momentum quadrature noise may then be given by: <MAT> <MAT> Explicit expressions for the probability density functions of ξq and ξp are provided in Appendix B herein. In the case where σ is much smaller than <MAT>, the reshaped ancilla quadrature noise <MAT> and <MAT> lie in the range <MAT> with a very high probability. Thus, <MAT> and <MAT>, and the logical position and momentum quadrature noise may be given by: <MAT> <MAT> That is, the variance of the logical position quadrature noise may be reduced by a factor of <NUM> by performing QEC using the two-mode GKP-repetition codes. This is due to the syndrome measurement of the reshaped ancilla position quadrature noise <MAT> which may be used to reduce the uncertainty of the data position quadrature noise <MAT>. Moreover, the variance of the logical momentum quadrature noise may remain unchanged despite the temporary increase <MAT> during the decoding procedure. Again, this may be due to the syndrome measurement of the reshaped ancilla momentum quadrature noise <MAT> <MAT> which may fully capture the transferred ancilla momentum quadrature noise <MAT> if <MAT>.

<FIG> is a plot of the standard deviations of the output logical quadrature noise, σq <NUM> and σp <NUM>, for the two-mode GKP-repetition code as a function of the input standard deviation, σ. The dashed line <NUM> represents <MAT> and the dashed line <NUM> represents σp = σ, respectively. The standard deviation of the output logical position quadrature noise <NUM> is reduced by a factor of <MAT> (e.g., <MAT>), while the standard deviation of the output logical momentum quadrature noise <NUM> remains unchanged (e.g., σp = σ) for σ ≲ <NUM>. Note that the condition σ ≲ <NUM> is translated to γ = σ<NUM> ≲ <NUM> in the case of pure excitation losses, where γ is the pure-loss probability. Thus, if the standard deviation of an additive Gaussian noise error is sufficiently small, the two-mode GKP-repetition coding scheme can reduce the noise of the position quadrature, while keeping the momentum quadrature noise unchanged. That is, the non-Gaussian GKP-repetition codes can correct additive Gaussian noise errors.

In an analogous two-mode Gaussian-repetition coding scheme, as described in Appendix A herein, the variance of the position quadrature noise is reduced by a factor of <NUM> (e.g., <MAT>) similarly as for the two-mode GKP-repetition coding scheme. However, the variance of the momentum quadrature noise is increased by the same factor (i.e., <MAT>). This demonstrates that, in the case of Gaussian-repetition codes, the position and momentum quadrature noises are only squeezed (σqσp = σ<NUM>) instead of being corrected (σqσp < σ<NUM>), reaffirming the previous no-go results for Gaussian QEC schemes.

An important difference between the two-mode GKP-repetition code and previous Gaussian repetition codes is that in the latter case the ancilla momentum quadrature noise that is transferred to the data mode (see <MAT> in <MAT>) is left completely undetected, whereas in the two-mode GKP-repetition code schema, the ancilla momentum quadrature noise is captured by measuring the ancilla momentum quadrature operator modulo fu. Such a limitation of the Gaussian-repetition code is a general feature of any Gaussian QEC scheme which relies on homodyne measurements of ancilla quadrature operators. With homodyne measurements, one can only monitor the noise in one quadrature of a bosonic mode, while the noise in the other conjugate quadrature is left completely undetected.

The canonical GKP state presents a work-around for this barrier as a non-Gaussian resource which allows simultaneous measurements of both the position and momentum quadrature operators modulo <MAT>. Such non-Gaussian modular simultaneous measurements of both position and momentum quadrature operators performed by using canonical GKP states are accordingly fundamentally different from Gaussian heterodyne measurements where both quadrature operators are measured simultaneously but in a noisy manner.

The two-mode GKP-repetition code may further be compared to the conventional three-bit repetition code (e.g., |<NUM>L〉 =|<NUM>〉 ⊗<NUM> and |<NUM>L〉 =|<NUM>〉 ⊗<NUM>) which can correct single bit-flip errors. First, in the GKP-repetition code, it is possible to use only two bosonic modes to reduce the variance of the position quadrature noise, whereas at least three qubits are needed to suppress qubit bit-flip errors in the case of multi-qubit repetition code. However, while the GKP-repetition code can be implemented in a more hardware-efficient way, it does not reduce the variance of the position quadrature noise quadratically but instead only reduce the variance by a constant factor (e.g., <MAT> if <MAT>). On the other hand, the three-qubit repetition code can reduce the bit-flip error probability quadratically from p to pL ≃ <NUM>p<NUM> if p « <NUM>.

Such a quadratic (or even higher order) suppression of Pauli errors is a significant step towards qubit-based, fault-tolerant universal quantum computation. Thus, it is desirable in oscillator encoding schema to have such a quadratic suppression of additive Gaussian noise errors in order to go beyond the reduction by a constant factor described above.

Below, a two-mode GKP-squeezing code is introduced and shown to suppress both the position and momentum quadrature noise quadratically up to a small logarithmic correction. In particular, the proposed scheme can achieve such a quadratic noise suppression by using only two bosonic modes, including one data mode and one ancilla mode, and therefore is hardware-efficient. For example, as compared to qubit-based QEC, at least five qubits and high-weight multi-qubit gate operations are used to suppress both bit-flip and phase-flip errors quadratically.

<FIG> illustrates an example of a two-mode bosonic system <NUM> performing entangling using a two-mode GKP-squeezing code and subject to Gaussian noise errors after entangling of the two modes. The two-mode GKP-squeezing code may be defined as: <MAT> where |ψ〉 = ∫ dqψ(q)|q<NUM> = q) is an arbitrary bosonic state of the first data mode, and |GKP〉 is the canonical GKP state of the second, ancilla mode. Further, TS<NUM>,<NUM>(G) is the two-mode squeezing operation <NUM> acting on the data and ancilla states with a gain of G ≥ <NUM>. In the Heisenberg regime, the two-mode squeezing operation TS<NUM>,<NUM>(G) transforms the quadrature operators x = <MAT> into <MAT>, where the <NUM> × <NUM> symplectic matrix STS(G) associated with TS<NUM>,<NUM>(G) is given by: <MAT> Here, I = diag(<NUM>,<NUM>) is the <NUM> × <NUM> identity matrix and Z = diag(<NUM>,-<NUM>) is the Pauli Z matrix.

In some embodiments, the two-mode squeezing operation TS<NUM>,<NUM>(G) <NUM> can be decomposed into a sequence of <NUM>:<NUM> beam splitter operations and single-mode squeezing operations, such as: <MAT> where <MAT>. Here, Sqk(λ) is the single-mode squeezing operation that transforms q̂k and p̂k into λq̂k and p̂k/λ, respectively. Also, BS<NUM>,<NUM>(η) is the beam splitter interaction between the modes <NUM> and <NUM> with a transmissivity η ∈ [<NUM>,<NUM>] and is associated with a <NUM> × <NUM> symplectic matrix represented as: <MAT> Note that the squeezing parameter λ (or the gain G) may be chosen at will to optimize the performance of the error correction scheme.

Upon the addition of independent and identically distributed Gaussian noise errors <NUM>, the quadrature operator x̂' may be further transformed into x̂" = x̂' + ξ, where ξ = <MAT> is the quadrature noise vector obeying <MAT>. <FIG> illustrates an example of the two-mode bosonic system <NUM> performing a decoding operation and error correction using a two-mode GKP-squeezing code. The inverse squeezing operation <NUM> (TS<NUM>,<NUM>(G))† may transform the quadrature operator into x̂‴ = (STS(G))-<NUM>x̂" = x̂ + z, where <MAT> is the reshaped quadrature noise vector which is given by: <MAT> As in the case of the GKP-repetition code, information about the reshaped ancilla quadrature noise <MAT> and <MAT> may be extracted by simultaneously measuring the position and momentum quadrature operators of the ancilla mode modulo <MAT> at points <NUM> and <NUM> of <FIG>. These modular measurement outcomes may then be used to estimate the reshaped data quadrature noise <MAT> and <MAT>.

The bare quadrature noise <MAT> is uncorrelated and thus its covariance matriz is proportional to the <NUM> × <NUM> identity matrix: Vξ = σ<NUM>diag(<NUM>,<NUM>,<NUM>,<NUM>). Additionally, the covariance matrix of the reshaped quadrature noise, z = (STS(G))-<NUM>ξ, may be given by: <MAT> Accordingly, the reshaped data and ancilla quadrature noise are correlated whenever G > <NUM>. That is, the inverse of the encoding circuit (TS(G))† may convert the uncorrelated quadrature noise ξ into a correlated noise z. In particular, in the limit of G » <NUM>, the covariance Vz may be asymptotically given by: <MAT> and therefore the data and ancilla position quadrature noise are anti-correlated (i.e., <MAT>), and the data and ancilla momentum quadrature noise are correlated (i.e., <MAT>). These correlations in the G » <NUM> limit allow us to reliably estimate the data quadrature noise <MAT> and <MAT> based solely on the knowledge of the ancilla quadrature noise <MAT> and <MAT>.

However, though a large gain G may be desirable to achieve such noise correlations, the gain G may not be increased indefinitely because then the reshaped ancilla quadrature noise <MAT> and <MAT> (whose variances are given by (<NUM>G - <NUM>)σ<NUM>) may not be contained within the distinguishable range, <MAT>. Therefore, the gain G may be chosen such that the reshaped ancilla quadrature noise lies mostly in this distinguishable range.

Based on the outcomes of the measurements <NUM>, <NUM> of the ancilla quadrature operators modulo <MAT>, the reshaped ancilla noise <MAT> and <MAT> can be estimated to be the smallest noise values that are compatible with the modular measurement outcomes such that <MAT> Then, the reshaped data position and momentum quadrature noise <MAT> and <MAT> may be further estimated to be <MAT> <MAT> The choice of these estimates is described in further detail in Appendix C herein. In the G » <NUM> limit, the above estimates <MAT> and <MAT> may be respectively reduced to <MAT> and <MAT>, which are reasonable because then the data and ancilla position quadrature noise may be anticorrelated whereas the data and ancilla momentum quadrature noise may be correlated.

Finally, the noise error in the data mode may be corrected by applying the counter displacement operations <MAT> and <MAT>, shown as elements <NUM> and <NUM> in <FIG>, respectively. The output logical quadrature noise may then be given by: <MAT> <MAT> As described in Appendix C herein, the probability density functions of the output logical quadrature noise ξq and ξp are identical to each other and are given by a mixture of Gaussian distributions: <MAT> Here, <MAT> is the probability density function of the Gaussian distribution <IMG>(<NUM>, σ<NUM>) and qn and µn are given by <MAT>.

Thus, the mixture probabilities, qn, sum up to unity, i.e., <MAT>, and qn represents the probability that the ancilla position quadrature noise <MAT>, whose standard deviation is given by <MAT>, lies in the range <MAT>. In this case, <MAT>, causing a misidentification of <MAT> by a finite shift <MAT>. Such a misidentification may result in an undesired shift, µn, to the data mode through a miscalibrated counter displacement.

The variance (σL)<NUM> of the output logical quadrature noise may now be considered. By using the probability density function, Q(ξ), the variance (σL)<NUM> may be described by: <MAT> The gain G may be chosen to optimize the performance of the two-mode GKP-squeezing code. Here, G may be chosen such that the standard deviation of the output logical quadrature noise σL is reduced.

<FIG> illustrates the standard deviations of the output logical quadrature noise, σq = <MAT>, as a function of the input standard deviation σ for the two-mode GKP-squeezing code in curve <NUM>. The dashed curve <NUM> represents <MAT>. Relatedly, <FIG> illustrates two-mode squeezing gain G* in curve <NUM> configured to achieve <MAT> as a function of the input standard deviation for the two-mode GKP-squeezing code. The curve <NUM> has been translated to single-mode squeezing in the units of decibel 20log<NUM>λ* where <MAT> <MAT>. The dashed curve <NUM> represents <MAT>.

For σ ≥ <NUM>, the gain G* is given by G* = <NUM> and accordingly the two-mode GKP-squeezing code may not reduce the noise in the data state: <MAT>. However, if the standard deviation of the input noise is small enough, e.g., σ < <NUM>, the gain G* may be strictly larger than <NUM> and the standard deviation of the output logical quadrature noise σL may be made smaller than the standard deviation of the input quadrature noise <MAT>. Note that the condition σ < <NUM> corresponds to γ = σ<NUM> < <NUM> for the pure excitation loss error where γ is the loss probability. Below, asymptotic expressions for G* and <MAT> in the σ « <NUM> limit are provided and the two-mode GKP-squeezing code is shown to quadratically suppress additive Gaussian noise errors up to a small logarithmic correction.

Useful solutions may be found in the regime where <MAT> holds and thus the ancilla quadrature noises <MAT> and <MAT> fall within the range <MAT> with a very high probability. In this regime, the variance, (σL)<NUM>, is approximately given by: <MAT> where <MAT> is the complementary error function. Further assuming σ « <NUM>, the variance (σL)<NUM> may be minimized when the gain G is given by: <MAT> which is derived in Appendix C. Then, the standard deviation of the output logical quadrature noise <MAT> may be given by: <MAT>.

As can be seen from <FIG>, these asymptotic expressions agree with the numerical results. In particular, the standard deviation of the logical quadrature noise <MAT> decreases quadratically as σ decreases (i.e., <MAT>) up to a small logarithmic correction. For example, when σ = <NUM> (which corresponds to γ = σ<NUM> = <NUM> for the pure excitation loss error), the gain may be given by G* = <NUM>, which may need 20log<NUM>λ* = <NUM>. 35dB single-mode squeezing operations. In this case, the resulting standard deviation of the output noise may be given by <MAT> <NUM> which corresponds to the loss probability <NUM>%. This corresponds to a QEC "gain" for the protocol of <NUM>/<NUM> ≃ <NUM> in terms of the loss probability and <NUM>/<NUM> ≃ <NUM> in terms of displacement errors.

The two-mode bosonic system <NUM>, using a two-mode GKP-squeezing code, is hardware-efficient because only two bosonic mode-one data mode and one ancilla mode-may be used to achieve the quadratic suppression of both momentum and position quadrature noise up to a small logarithmic correction. In contrast, in the case of multi-qubit QEC, at least five qubits and high-weight multi-qubit operations are needed to suppress both the bit-flip and phase-flip errors quadratically. While the two-mode GKP-squeezing code is relatively simple and achieves a quadratic noise suppression, it may be desirable to achieve higher order noise suppression. Below, having this in mind, the non-Gaussian quantum error correction codes are generalized to an even broader class of non-Gaussian quantum error-correcting codes.

The two-mode GKP-repetition code and the two-mode GKP-squeezing code may be generalized to an even broader class of non-Gaussian oscillator codes, or, GKP-stabilizer codes. Such GKP-stabilizer codes may be defined generally such that M oscillator modes are encoded into N oscillator modes as: <MAT> where |Ψ〉 is an arbitrary M-mode bosonic state in the first M modes (data modes) and |GKP〉⊗N-M is the N - M copies of the canonical GKP states in the last N - M modes (ancilla modes) and <MAT> is an encoding Gaussian unitary operation. For example, the two-mode GKP-repetition code may be expressed generally as M = <NUM>, N = <NUM>, and <MAT>. Similarly, the two-mode GKP-squeezing code may be expressed generally as M = <NUM>, N = <NUM>, and <MAT>.

Since one can choose any Gaussian operation as an encoding circuit <MAT>, this GKP-stabilizer formalism is at least as flexible as the stabilizer formalism for qubit-based QEC which encompasses almost all conventional multi-qubit QEC schemes. Therefore, this GKP-stabilizer formalism has a great deal of potential. As an illustration, in Appendix D, we introduce a family of GKP-stabilizer codes, GKP-squeezed-repetition codes, and show that the N-mode GKP-squeezed-repetition code can suppress additive Gaussian noise errors to the Nth order, as opposed to the second order for the two-mode GKP-squeezing code. In addition to the GKP-squeezed-repetition codes, there may be many other useful families of GKP-stabilizer codes.

<FIG> illustrates an example of a generalized encoding circuit <NUM> for use with a GKP-stabilizer code and a general logic Gaussian operator <MAT>. The data states, |Ψ〉, and ancilla states, |GKP〉, may be encoded (e.g., may be entangled) using an encoding operation <MAT> in block <NUM>. The states may then be disentangled in block <NUM> using an inverse encoding operation <MAT>. A logical Gaussian operator <MAT> may be applied to one or more of the data states in block <NUM>, and an auxiliary Gaussian operator <MAT> may be applied to one or more of the ancilla states in block <NUM>. A final encoding operation <MAT> may be performed on the data and ancilla states in block <NUM> to correlate the results of the logical Gaussian operator <MAT> with the information stored in the ancilla states.

In general, for any GKP-stabilizer code with a Gaussian encoding circuit <MAT>, a logical Gaussian operation <MAT> may be implemented by using physical Gaussian operations, such as: <MAT>, where <MAT> is the Gaussian operation to be applied to the data modes and <MAT> is an auxiliary Gaussian operation to the ancilla modes (see e.g., <FIG>). Physical Gaussian operations may be sufficient for the implementation of logical Gaussian operations because the input ancilla GKP state |GKP〉⊗N-M is the only non-Gaussian resource in the encoding scheme and the remaining circuit <MAT> is also Gaussian. Auxiliary Gaussian operations may be used to simplify the form of the physical Gaussian operation <MAT>.

Many interesting continuous variable quantum information processing tasks such as the KLM protocol, boson sampling, and simulation of vibrational quantum dynamics of molecules use Gaussian operations and, as non-Gaussian resources, input single- or multi-photon Fock states and photon number measurements of the output modes. Since the non-Gaussian resources are consumed locally only in the beginning and the end of the computation, any imperfections in these non-Gaussian resources may be addressed independently from the imperfections in the Gaussian operations. Below, improving the quality of Gaussian operations such as beam splitter interactions is described.

Returning to the two-mode GKP-squeezing code introduced previously herein. For simplicity, consider two logical oscillator modes encoded in four physical oscillator modes via the encoding Gaussian circuit <MAT>. Here, the modes <NUM>, <NUM> are the data modes and the modes <NUM>, <NUM> are the ancilla modes which are used for syndrome error detection. Note that the encoding Gaussian circuit of the GKP-two-mode-squeezing code has the following property: <MAT> where <MAT>. That is, by choosing <MAT>, the logical beam splitter interaction can be implemented transversally by applying a pair of beam splitter interactions BS<NUM>,<NUM>(η) and BS<NUM>,<NUM>(η) to the data modes and the ancilla mode, as shown in <FIG>.

<FIG> illustrates an example of a multi-mode bosonic system <NUM> performing a logical beam splitter operation in the absence of Gaussian noise error during the beam splitter interaction. Data states and ancilla states, initialized in GKP states, may be entangled by squeezing operations <NUM> and <NUM> prior to the logical beam splitter operations <NUM> and <NUM>. Thereafter, the data and ancilla states may be disentangled by inverse squeezing operations <NUM> and <NUM>.

The transversality in the equation describing BS<NUM>,<NUM>(η)BS<NUM>,<NUM>(η) above may be used to correct additive Gaussian noise errors that occur during the implementation of a beam splitter interaction. Note that any passive beam splitter interactions commute with iid additive Gaussian noise errors (see Appendix E). Therefore, the beam splitter interaction BS<NUM>,<NUM>(η)BS<NUM>,<NUM>(η) that are continuously corrupted by iid additive Gaussian noise errors can be understood as the ideal beam splitter interaction BS<NUM>,<NUM>(η)BS<NUM>,<NUM>(η) followed by an iid additive Gaussian noise channel <MAT> where the variance of the additive noise σ<NUM> is proportional to the time needed to complete the beam splitter interaction.

<FIG> illustrates an example of a multi-mode bosonic system <NUM> performing a logical beam splitter operation in the presence of Gaussian noise error during the beam splitter interaction. In some embodiments, the data states and ancilla states, initialized in GKP states, may be entangled by squeezing operations <NUM> and <NUM> prior to the logical beam splitter interactions <NUM> and <NUM>. The data and ancilla states may then be subject to uncorrelated Gaussian noise <NUM>. Thereafter, the data and ancilla states may be disentangled using inverse squeezing operations <NUM> and <NUM>. Such a system may be more simply described by the system shown on the right, wherein data states and ancilla states, initialized as GKP states, may be subject to the logical beam splitter interactions <NUM> and <NUM> and then subject to correlated Gaussian noise <NUM> and <NUM>.

As shown in <FIG>, the uncorrelated additive Gaussian noise error that has been accumulated during the logical beam splitter interaction may be converted into a correlated additive Gaussian noise error through the inverse of the encoding circuit <MAT> TS<NUM>,<NUM>(G)†TS<NUM>,<NUM>(G)† in the decoding procedure. That is, after the noise reshaping, the quadrature noise in the data modes <NUM> and <NUM> are correlated with the ones in the ancilla modes <NUM> and <NUM>, respectively. Because of the correlation between the data modes and the ancilla modes, information about the reshaped data quadrature noise <MAT> and <MAT> may be determined by extracting information about the reshaped ancilla quadrature noise <MAT> and <MAT>. This information may then be used to suppress the additive Gaussian noise accumulated in the data mode quadratically up to a small logarithmic correction.

Note that so far ideal GKP states have been assumed to clearly demonstrate the error-correcting capability of GKP-stabilizer codes. In the following discussion, examples of experimental realization of GKP-stabilizer codes are described. Especially, issues related to the use of realistic noisy GKP states are discussed, as they may be an important factor in the design of practical GKP-stabilizer codes.

Experimental realization of GKP-stabilizer codes and the effects of realistic imperfections are discussed herein. Recall that the non-Gaussian resource used for implementing GKP-stabilizer codes is the preparation of a canonical GKP state. While Gaussian resources are readily available in many realistic bosonic systems, preparing a canonical GKP state is not strictly possible because it would require infinite squeezing of the state. Recently, however, finitely-squeezed approximate GKP states have been realized in a trapped ion system by using a preparation scheme with post-selection and in a circuit QED system by using a deterministic scheme. Thus, the two-mode GKP-squeezing code may, in principle, be implemented in the state-of-the-art quantum computing platforms using approximate GKP states.

Imperfections in realistic GKP states, such as finite squeezing, may add additional quadrature noise to the quantum information system. Therefore, the performance of the two-mode GKP-squeezing code will be mainly limited by the finite squeezing of the approximate GKP states. Indeed, a non-trivial QEC gain <MAT>, as shown below, may be achievable with the two-mode GKP-squeezing code when the supplied GKP states have a squeezing larger than <NUM>.

The squeezing achieved thus far in experimentally realized GKP states ranges from <NUM>. 5dB to <NUM>. In this regard, the oscillator encoding schemes discussed herein are compatible with non-deterministic GKP state preparation schemes because the required GKP states can be prepared offline and then supplied to the error correction circuit in the middle of the decoding procedure. It may therefore be advantageous to sacrifice the probability of success of the GKP state preparation schemes to aim to prepare a GKP state of higher quality (e.g., with a squeezing value larger than <NUM>. 0dB) by using post-selection.

In general, the imperfections in approximate GKP states may be especially detrimental to a GKP-stabilizer code involving a large squeezing parameter because noise and other imperfections may be significantly amplified by the large squeezing operations. With this concern in mind, turn back to the two-mode GKP-squeezing code and recall that the gain G* may be asymptotically given by G* ∝ <NUM>/σ<NUM> in the σ « <NUM> limit. If the standard deviation of the input noise is very small, then a very large gain parameter G* » <NUM> (or <MAT>). However, the two-mode GKP-squeezing code has been designed so that any imperfections in approximate GKP states may not be amplified by the large squeezing operations.

Recall that an approximate GKP state with a finite squeezing can be modeled by |GKPΔ〉 ∝ exp [-Δn̂]|GKP〉. The finitely-squeezed GKP state |GKPΔ〉 may be converted via a noise twirling into <MAT> That is, for example, an ideal canonical GKP state may be corrupted by an incoherent random shift error, <IMG>[σgkp]. Here, <MAT> is the variance of the additive noise associated with the finite GKP squeezing. The noise standard deviation <MAT> characterizes the width of each peak in the Wigner function of an approximate GKP state. The GKP squeezing is then defined as <MAT>. The GKP squeezing sgkp quantifies how much an approximate GKP state is squeezed in both the position and the momentum quadrature in comparison to the vacuum noise variance <NUM>/<NUM>.

<FIG> present a multi-mode bosonic system <NUM> for implementation of the two-mode GKP-squeezing code. <FIG> illustrates an example of a multi-mode bosonic system <NUM> performing entangling using a two-mode GKP-squeezing code and subject to Gaussian noise errors before and after entangling of the two modes. The ancilla state may be subject to a Gaussian noise error <NUM> before application of the squeezing operation <NUM> to entangle the data and ancilla states. After the squeezing operation, the entangled states may be subject to a further Gaussian noise error <NUM>.

<FIG> illustrates an example of a multi-mode bosonic system <NUM> performing disentangling using a two-mode GKP-squeezing code and performing error correction using an additional measurement ancilla mode. In some embodiments, the data and ancilla states may be disentangled using an inverse squeezing operation <NUM>. An additional measurement ancilla state, initialized as a GKP state, may be introduced in the process of <FIG> to simultaneously measure the position and momentum operators of the ancilla mode modulo <MAT>. The measurement ancilla state may be subject to additional Gaussian errors <NUM> and then may be entangled with the ancilla state through a SUM gate <NUM>. Thereafter, the position and momentum operators of the ancilla state may be measured by measurement <NUM> of the ancilla state and measurement <NUM> of the measurement ancilla state. The measured momentum and position operators may then be applied in the form of displacement operations <NUM> and <NUM> to correct the Gaussian errors in the data mode.

Note that, in the processes of <FIG>, one GKP state may be consumed to perform the simultaneous and modular position and momentum measurements. In comparison, using the measurement circuits of <FIG> may consume two GKP states for the simultaneous modular quadrature measurements because the measurement circuits of <FIG> are configured for performing non-destructive measurements. While the first measurement of <FIG> (e.g., the modular position measurement) may be performed in a non-destructive manner, the following measurement (e.g., the modular momentum measurement) may be performed in a destructive manner since the quantum state in the ancilla mode is no longer needed. Rather, the classical measurement outcomes <MAT> and <MAT> desired. This is the reason why, in <FIG>, the momentum quadrature of the second mode (e.g., the ancilla mode) may be measured in a destructive way after the modular position measurement. Saving such non-Gaussian resources in this manner is particularly useful in the regime where the finite squeezing of an approximate GKP state is a limiting factor.

Because of such non-Gaussian resource saving described above, two GKP states may be supplied to implement the two-mode GKP-squeezing code, in some embodiments. One of the supplied GKP states may be used as the input of the ancilla mode and the other supplied GKP state may be for the simultaneous and modular ancilla quadrature measurements. Assuming that these two supplied GKP states are corrupted by an additive Gaussian noise channel <IMG>[σgkp], i.e. <MAT>, as shown in <FIG>. Due to this additional noise associated with the finite squeezing of the GKP states, the estimated reshaped ancilla quadrature noise may be corrupted as follows: <MAT> <MAT> Here, <MAT> and <MAT> are the additional noises caused by the finite GKP squeezing, and they follow <MAT>. Such additional noise may then be propagated to the data mode through the miscalibrated counter displacement operations based on noisy estimates. In the presence of additional GKP noise, the sizes of the counter displacements <MAT> and <MAT> are given by <MAT> <MAT> and do not explicitly depend on G in the limit of G » <NUM>. Therefore, the additional GKP noise <MAT> and <MAT> may be added to the data quadrature operators without being amplified by the large gain parameter G » <NUM>. This absence of the noise amplification is an important feature of the proposed error correction scheme described herein and is generally not the case for a generic GKP-stabilizer code involving large squeezing operations (see e.g., Appendix D for an illustration).

In Appendix C, an analysis of the adverse effects of the finitely-squeezed GKP states is provided. In particular, the variance of the output logical quadrature noise (σL)<NUM> is derived as a function of the input noise standard deviation σ, the GKP noise standard deviation σgkp, and the gain of the two-mode squeezing G. As described previously herein, the gain G of the two-mode squeezing is optimized to minimize the output logical noise standard deviation σL for given σ and σgkp.

<FIG> plots QEC gain, <MAT>, as a function of the input noise standard deviation for different values of GKP squeezing including: <NUM>. 9dB in curve <NUM> (not visible), <NUM>. 0dB in curve <NUM>, <NUM>. 0dB in curve <NUM>, 15dB in curve <NUM>, 20dB in curve <NUM>, 25dB in curve <NUM>, and 30dB in curve <NUM>. QEC gain for a GKP state with infinite squeezing is plotted in dashed curve <NUM> for comparison.

<FIG> plots gain, G*, of the two-mode squeezing as a function of input noise standard deviation for different values of GKP squeezing including: <NUM>. 9dB in curve <NUM> (not visible), <NUM>. 0dB in curve <NUM>, <NUM>. 0dB in curve <NUM>, 15dB in curve <NUM>, 20dB in curve <NUM>, 25dB in curve <NUM>, and 30dB in curve <NUM>, in accordance with some embodiments described herein. The gain of the two-mode squeezing for a GKP state with infinite squeezing is plotted in dashed curve <NUM> for comparison.

The nontrivial QEC gain <MAT> is observed to be achieved when the supplied GKP states have a squeezing larger than the critical value <NUM>. Also, when the supplied GKP states have a squeezing of 30dB, the QEC gain is given by <MAT>, which is achieved when σ = <NUM>. For comparison, the QEC gain at the same input noise standard deviation σ = <NUM> is given by <MAT> when the ideal canonical GKP states are used to implement the two-mode GKP-squeezing code. These two values (<NUM> and <NUM>) being of the same magnitude indicates that the additional GKP noise is not significantly amplified by the large (e.g., <NUM>. 3dB) single-mode squeezing operations used in this regime.

Here, the N-mode Gaussian-repetition code is introduced and analyzed, which is a generalization of the three-mode Gaussian-repetition code. In particular, we introduce the maximum likelihood estimation of the data position quadrature noise <MAT> for the N-mode Gaussian-repetition code, which is a motivator behind our choice of the estimate <MAT> in connection with the two-mode GKP-repetition code described herein.

Consider an arbitrary oscillator state [ψ〉 = ∫ dqψ(q)|q̂<NUM> = q〉. The state [ψ〉 can be embedded in N oscillator modes via the N-mode Gaussian-repetition code as follows: <MAT> where the first mode is the data mode and the rest are the ancilla modes. Note that the data position eigenstate |q̂<NUM> = q〉 is mapped into <MAT> through the encoding procedure. As shown in <FIG>, this encoding can be realized by applying a sequence of Gaussian SUM gates SUM<NUM>→k ≡ exp[-iq̂jp̂k] where k ∈ {<NUM>, ··· , N}. Upon the encoding circuit, the quadrature operators are transformed into <MAT> <MAT> <MAT> <MAT> where k ∈ {<NUM>, ··· , N}. The oscillator modes may be assumed to undergo independent and identically distributed additive Gaussian noise errors <MAT>, e.g., <MAT> and <MAT>. The added noise <MAT> follow an independent and identically distributed Gaussian random distribution <MAT>.

<FIG> illustrates an example of multi-mode bosonic system <NUM> performing QEC using an N-mode Gaussian repetition code.

The multi-mode bosonic system <NUM> includes a data state and N ancilla states. The N ancilla states may be entangled with the data state through N SUM gates <NUM>, after which the data and ancilla states may be subject to correlated Gaussian noise errors <NUM>. To disentangle (e.g., to decode) the data and ancilla states, N inverse SUM gates <NUM> may be applied, after which position quadrature operators of the ancilla states may be measured using N measurements <NUM>. In some embodiments, the measurements <NUM> may be homodyne measurements. The data state may be error corrected using displacement operator <NUM> based on the position quadrature operator measurements <NUM>.

The goal of the decoding procedure depicted in <FIG> is to extract information about the added noise <MAT> and <MAT> through a set of syndrome measurements. The decoding procedure begins with the inverse of the encoding circuit, i.e., by a sequence of inverse-SUM gates <MAT> for k ∈ {<NUM>, ··· , N}. Upon the inverse of the encoding circuit, the quadrature operators may be transformed into <MAT> and <MAT>, where the reshaped quadrature noise is given by <MAT> <MAT> <MAT> <MAT> where k ∈ {<NUM>, ··· , N}. Then, by performing measurements <NUM> of the ancilla position quadrature operators, the values of <MAT> can be extracted for all k ∈ {<NUM>, ··· , N}. This is because the ancilla modes may be initially in the position eigenstates |q̂k = <NUM>〉 and thus measuring <MAT> is equivalent to measuring <MAT> for all k ∈ {<NUM>, ··· , N}. Note, however, that information about the reshaped momentum quadrature noise <MAT> may not be extracted, which is an important limitation of Gaussian-repetition codes.

From the extracted values of <MAT>, the position quadrature noise may be inferred to be given by <MAT>. Then, the undetermined data position quadrature noise <MAT> may be estimated using a maximum likelihood estimation. Since noise errors with smaller <MAT> are more likely, <MAT> may be estimated as: <MAT> from the syndrome measurement outcomes <MAT>. Note that for N = <NUM>, this equation reduces to <MAT>, which is why <MAT> may be chosen as an estimate of <MAT>. Finally, the decoding procedure may end with an application of the counter displacement operation <MAT> <NUM> to the data oscillator mode, as shown in the example of <FIG>.

As a result of the encoding and decoding procedures, a logical additive noise error q̂<NUM> → q̂<NUM> + ξq and p̂<NUM> → p̂<NUM> + ξp may be present in the data oscillator mode, where the output logical position and momentum quadrature noise are given by <MAT> <MAT> Because <MAT>, these expressions may be rewritten as: <MAT> <MAT> Thus, the variance of the position quadrature noise is reduced by a factor of N, but the variance of the momentum quadrature noise is increased by the same factor. The latter increase is due to the transfer of ancilla momentum quadrature noise to the data momentum quadrature (see <MAT> in <MAT>) which is left undetected by the position homodyne measurements. As a result, the product of the noise standard deviations remains unchanged at the end of the error correction procedure (e.g., σqσp = σ<NUM>). This implies that Gaussian-repetition codes can only squeeze the Gaussian quadrature noise but cannot actually correct them. This reaffirms the previous no-go results for correcting Gaussian errors with Gaussian resources.

Here, explicit expressions for the probability density functions of the logical quadrature noise ξq and ξp for the two-mode GKP-repetition code are provided. Recall that: <MAT> <MAT> and note that the logical quadrature noise ξq and ξp for the two-mode GKP-repetition are given by: <MAT> <MAT> where <MAT>, Rs(z) ≡ z - n*(z)s, and <MAT> ns|. Let pσ(z) denote the probability density function of a Gaussian distribution with zero mean and variance σ<NUM>, i.e., <MAT>. Then, the probability density functions Q(ξq) and P(ξp) of the logical quadrature noise ξq and ξp may be given by: <MAT> where δ(x) is the Dirac delta function. Note that Rs(z) can be expressed as: <MAT> where I{C} is an indicator function, e.g., I{C} = <NUM> if C is true I{C} = <NUM> if C is false. Then, using this expression for Rs(z), Q(ξq) and P(ξp) can be written in more explicit forms as follows: <MAT> <MAT>.

<FIG> as described herein may be obtained by numerically computing these probability density functions and then evaluating the standard deviations of the obtained numerical probability density functions.

Here, the underlying reasons behind the choice of the estimates <MAT> and <MAT> for the GKP-two-mode-squeezing code are explained. Recall that the covariance matrix of the reshaped noise <MAT> is given by <MAT> For now, ignore that <MAT> and <MAT> can be measured only modulo <MAT> and instead assume that their exact values may be known. Note that <MAT> is correlated with <MAT>, whereas <MAT> is correlated with <MAT>. Consider the estimates of the form <MAT> and <MAT>, where cq and cp are constants. The values of cq and cp may be chosen such that the variances of <MAT> and <MAT> are minimized, as <MAT> and <MAT> are given by <MAT> <MAT> they are minimized when <MAT> <MAT> Therefore, if both <MAT> and <MAT> are precisely known, the estimates of <MAT> and <MAT> are given by <MAT> <MAT> However, as <MAT> and <MAT> may be measured modulo <MAT>, if <MAT> and <MAT> are replaced by <MAT> <MAT> and <MAT> the estimates <MAT> and <MAT> as described herein may be determined.

An explicit expression for the probability density functions of the logical quadrature noise ξq and ξp for the two-mode GKP-squeezing code is provided herein. Recall that: <MAT> <MAT> where <MAT> follows a Gaussian distribution with zero means and the covariance matrix Vz. By using the expression for Rs(z) of Appendix B, the probability density functions of the quadrature noise may be given by <MAT> and similarly <MAT> where qn(= q-n) and µn are defined as <MAT> <MAT>.

Finally, the asymptotic expressions for the gain G* and the standard deviation <MAT> are derived. Recall that assuming <MAT> gives: <MAT> Assuming further that G » <NUM> (which is relevant when σ « <NUM>) and using the asymptotic formula for the complementary error function, e.g., <MAT> the above expression for σL may be simplified as <MAT> where x ≡ <NUM>/(<NUM>G - <NUM>). The optimum x* can be found by solving f'(x*) = <NUM>. Note that f'(x*) is given by <MAT> Thus, x* should satisfy <MAT> where the first equation has been iteratively entered into itself to arrive at the second equation. Since loge(···) » <NUM>, the second term in the second line may be disregarded. By further neglecting the logarithmic factor <MAT>, x* may be given by <MAT> Since <MAT>, the following equation follows: <MAT> The value <MAT> is then approximately given by <MAT> Since <MAT>, the second term may be disregarded to obtain: <MAT>.

Now consider the case with noisy GKP states (see <FIG> and related description herein). Then, <MAT> where the GKP noise <MAT> is independent of <MAT> and <MAT> and follows <MAT>. Then, the probability density function Q(ξq) is given by <MAT> Thus, the variance of the output logical quadrature noise <MAT> is given by: <MAT>.

A family of GKP-stabilizer codes, called GKP-squeezed-repetition codes herein, are introduced in this section. These N-mode GKP-squeezed-repetition codes can suppress additive Gaussian noise errors to the Nth order, e.g., σq, σp ∝ σN, where σ is the standard deviation of the input additive Gaussian noise and σq and σp are the standard deviations of the output logical position and momentum quadrature noise, respectively.

Let [ψ〉 = ∫ dqψ(q)|q̂<NUM> = q〉 be an arbitrary one-mode bosonic state. The encoded logical state |ψL〉 of the N-mode GKP-squeezed-repetition code can be defined as: <MAT> where the encoding Gaussian circuit <MAT> is recursively defined as: <MAT> for N ≥ <NUM>. <FIG> illustrates an example of such an N-mode GKP-squeezed repetition code 1400a for N ≥ <NUM>. The N-mode GKP-squeezed repetition code 1400a may be performed by a system with a data state and a number of ancilla states (e.g., N - <NUM> ancilla states). The N-mode GKP-squeezed repetition code 1400a may be broken into several operations, <NUM> through <NUM>. These operations include several symplectic matrices <NUM>, <NUM>, and <NUM>, a SUM gate <NUM>, and a further squeezing operation <NUM> applied to the ancilla states only.

<FIG> illustrates an example of an N-mode GKP-squeezed repetition code 1400b for the base case N = <NUM>. The N-mode GKP-squeezed repetition code 1400b may be performed by a system with one data state and one ancilla state, and may be broken down into a SUM gate <NUM> and two symplectic matrices <NUM> and <NUM>. For this N-mode GKP-squeezed repetition code 1400b, <MAT> is given by: <MAT> and the symplectic matrix <MAT> associated with the encoding Gaussian circuit <MAT> is given by : <MAT>.

<FIG> illustrates an example of an N-mode GKP-squeezed repetition code 1400c for the case where N = <NUM>. The N-mode GKP-squeezed repetition code 1400c may be performed by a system with one data state and two ancilla states, and may be broken down into SUM gates <NUM> and <NUM> and symplectic matrices <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>. For this case, the symplectic matrix <MAT> with the encoding Gaussian circuit <MAT> is explicitly given by: <MAT> Thus, the reshaped quadrature noise vector <MAT> is given by <MAT> where <MAT> is the original quadrature noise vector whose elements follow independent and identically distributed Gaussian random distributions with variance σ<NUM>.

Through the noise reshaping, the data position quadrature noise is amplified by a factor of λ<NUM> (i.e., <MAT>). Note that, by choosing <MAT> with a small constant c « <NUM>, the ancilla position quadrature noise <MAT> and <MAT> can be contained within the unambiguously distinguishable range <MAT> with a very high probability. Then, by measuring the reshaped ancilla position quadrature noise of the second mode <MAT>, (modulo <MAT>) the amplified position quadrature noise <MAT> can be determined up to an error of <MAT>. Then, by further measuring the ancilla position quadrature noise of the third mode, <MAT> <MAT>, (modulo <MAT>) the residual error <MAT> can be determined up to an even smaller error of <MAT>. As a result, the position quadrature noise of the data mode may be reduced by a factor of λ<NUM> despite the temporary increase by the same factor before the correction. Since λ ∝ <NUM>/σ, this implies that the standard deviation of the position quadrature noise σq may be suppressed cubically, i.e., σq = σ/λ<NUM> ∝ σ<NUM>.

Additionally, through the noise reshaping, the data momentum quadrature noise may be immediately reduced by a factor of λ<NUM> (see <MAT> in <MAT>) but there may be transferred ancilla momentum quadrature noise (see <MAT> and <MAT> in <MAT>). Note that by measuring the ancilla momentum quadrature noise of the third mode, <MAT> (which is contained within the range <MAT> with a very high probability), the value of <MAT> may be extracted so that the transferred ancilla momentum quadrature noise <MAT> in <MAT> and also <MAT> in <MAT> may be eliminated. Then, by further measuring the reshaped ancilla momentum quadrature noise of the second mode (after eliminating <MAT>), the value of <MAT> may be extracted in order to eliminate the transferred ancilla momentum quadrature noise <MAT> in <MAT>. As a result, the reduced date momentum quadrature noise <MAT> is provided without any transferred ancilla momentum quadrature noise. Similarly as above, since λ ∝ <NUM>/σ, the momentum quadrature noise may be suppressed cubically, i.e., σp = σ/λ<NUM> ∝ σ<NUM>.

Generalizing these arguments, the N-mode GKP-squeezed-repetition code may be shown inductively to be able to suppress additive quadrature noise errors to the Nth order, e.g., σq, σp = σ/λN-<NUM> ∝ σN for any N ≥ <NUM>. These codes, however, may be susceptible to the realistic noise in GKP states (except for the N = <NUM> case), because the GKP noise can be amplified by large squeezing operations. For example, in the case of the three-mode GKP-squeezed-repetition code 1400c discussed above, a factor of λ may be multiplied into the obtained measurement outcome of <MAT> to eliminate the transferred noise <MAT> in the reshaped data momentum quadrature noise <MAT>. Therefore, the GKP noise that corrupts the measurement outcome of <MAT> may be amplified by a large squeezing parameter <MAT> when the GKP noise is propagated to the data mode via miscalibrated counter displacement operations.

Any passive beam splitter interactions are shown herein to commute with iid additive Gaussian noise errors. Consider n bosonic modes described by bosonic annihilation operators â<NUM>, ··· , ân. A general passive beam splitter interaction among these n modes is generated by a Hamiltonian of the following form: <MAT> where <MAT> for all k, l ∈ {<NUM>, ··· , n}. Note that the total excitation number is conserved under a general beam splitter interaction: <MAT>. Also, iid additive Gaussian noise errors are generated by the Lindbladian: <MAT> where <MAT>. Specifically, the Lindbladian generator <IMG> is related to the additive Gaussian noise error <IMG>[σ] by the relation: <MAT> where D is the diffusion rate and t is the time elapsed.

Having introduced the generators of the beam splitter interactions and iid additive Gaussian noise errors, the following commutation relation can be proven: <MAT> where <IMG>(p̂) ≡ -i[ĤBS,p̂] is the Lindbladian superoperator associated with a beam splitter Hamiltonian <MAT>. Note that for a general <IMG> = -i[Ĥ,•] and <IMG>[Â] that: <MAT> and similarly: <MAT> Therefore: <MAT> Using this general relation, one may find that: <MAT> Similarly: <MAT> Then, since <MAT>, [<IMG>, <IMG>] = <NUM> follows: <MAT>.

Since the generators of a beam splitter interaction and an iid additive Gaussian noise error commute with each other at the superoperator level, the result is: <MAT> where σ<NUM> = Dt and <MAT> and ÛBS = exp[-iĤBSt] is the desired n-mode beam splitter unitary operation. This implies that the noisy beam splitter interaction continuously corrupted by iid additive noise errors can be understood as the desired noiseless beam splitter interaction followed by an iid additive Gaussian noise channel with variance σ<NUM> = Dt, where D is the diffusion rate and t is the time needed to complete the beam splitter interaction.

Having thus described several aspects of at least one embodiment of this technology, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art.

Various aspects of the present technology may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its applications to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings.

The above-described embodiments of the technology described herein can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art by names such as CPU chips, GPU chips, microprocessor, microcontroller, or co-processor. Alternatively, a processor may be implemented in custom circuitry, such as an ASIC, or semi-custom circuitry resulting from configuring a programmable logic device. As yet a further alternative, a processor may be a portion of a larger circuit or semiconductor device, whether commercially available, semi-custom or custom. As a specific example, some commercially available microprocessors have multiple cores such that one or a subset of those cores may constitute a processor. Though, a processor may be implemented using circuitry in any suitable format.

Also, the various methods or processes outlined herein may be coded as software that is executable on one or more processors running any one of a variety of operating systems or platforms. Such software may be written using any of a number of suitable programming languages and/or programming tools, including scripting languages and/or scripting tools. In some instances, such software may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. Additionally, or alternatively, such software may be interpreted.

The techniques disclosed herein may be embodied as a non-transitory computer-readable medium (or multiple computer-readable media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory, tangible computer storage medium) encoded with one or more programs that, when executed on one or more processors, perform methods that implement the various embodiments of the present disclosure described above. The computer-readable medium or media may be transportable, such that the program or programs stored thereon may be loaded onto one or more different computers or other processors to implement various aspects of the present disclosure as described above.

The terms "program" or "software" are used herein to refer to any type of computer code or set of computer-executable instructions that may be employed to program one or more processors to implement various aspects of the present disclosure as described above. Moreover, it should be appreciated that according to one aspect of this embodiment, one or more computer programs that, when executed, perform methods of the present disclosure need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present disclosure.

Claim 1:
A quantum information device (<NUM>), comprising:
a plurality of quantum mechanical oscillators, comprising:
a first data quantum mechanical oscillator (<NUM>) configured to store information representing a first data state; and
a first ancilla quantum mechanical oscillator (<NUM>) coupled to the first data quantum mechanical oscillator, the first ancilla quantum mechanical oscillator configured to store information representing a first ancilla state, wherein the first ancilla state is initialized as a first Gottesman-Kitaev-Preskill state;
at least one electromagnetic radiation source (<NUM>) coupled to the plurality of quantum mechanical oscillators and configured to apply one or more drive waveforms to one or more of the first data quantum mechanical oscillator (<NUM>) and the first ancilla quantum mechanical oscillator (<NUM>), the one or more drive waveforms being configured to sequentially entangle and disentangle the information stored in the first data quantum mechanical oscillator (<NUM>) and the information stored in the first ancilla quantum mechanical oscillator (<NUM>); and
at least one measurement device (<NUM>) coupled to the first ancilla quantum mechanical oscillator (<NUM>), the at least one measurement device (<NUM>) configured to measure a position quadrature operator and a momentum quadrature operator of the first ancilla quantum mechanical oscillator (<NUM>), wherein:
the one or more drive waveforms further comprise displacement drive waveforms applied to the first data quantum mechanical oscillator (<NUM>) to correct an error in the information stored in the first data quantum mechanical oscillator (<NUM>), the displacement drive waveforms being determined using the measured position and momentum quadrature operators of the first ancilla quantum mechanical oscillator (<NUM>).