Photonic Quantum Computing

WOMANIUM Global Quantum Media Project Initiative — Winner of Global Quantum Media Project

Table of Contents:

01) Quantum Computing Models
02) Quantum Computing Platforms
03) Photonic Quantum Computing — Two Approaches
04) Computation via Teleportation — The Fundamental Ingredient
05) The EPR State Generator: Squeezed Light Sources
06) Brief Overview of the Detection System: Homodyne Detector
07) Graphical Representation of EPR and Cluster States
08) Measurement-Based Quantum Computation
09) Cluster State Generation
10) 2D Cluster State Generation
11) Implementing Gates
12) Noise Addition and Error Correction
13) GKP Qubits
14) Bosonic Error Correction
15) Architecture and Fault-Tolerant Threshold
16) Challange: Generation of error-correctable state — GKP state
17) Pros of Photonic Quantum Computing

Introduction

Photonic quantum computing holds great promise as an advanced method for quantum information processing, utilizing photons for qubit encoding and manipulation. Photons offer inherent resilience against decoherence and noise, making them ideal for scalable and fault-tolerant quantum computation. By leveraging sophisticated quantum optics techniques, photonic quantum computing aims to execute quantum algorithms and protocols with improved efficiency. However, a significant challenge lies in generating specialized states like the Gottesman-Kitaev-Preskill (GKP) state, crucial for error correction and achieving fault tolerance. Despite being in the early stages of development, photonic quantum computing shows realistic potential, but it faces challenges, such as GKP state generation.

Quantum Computing Models

Quantum Computing Platforms

Quantum computing can be realized in various platforms, each presenting unique advantages and challenges. Some of the leading platforms include:

1. Superconducting Qubits: Utilizes superconducting circuits to create stable quantum bits (qubits) for quantum computing.
2. Semiconductor Spins: Controls the quantum states of electron spins in semiconductor materials to perform quantum operations.
3. Nitrogen-Vacancy Centers: Manipulates the quantum states of nitrogen-vacancy defects in diamond materials for quantum computation.
4. Trapped Ions: Uses laser-trapped ions to create long-lasting qubits capable of high-fidelity quantum operations.
5. Neutral Atoms: Employs neutral atoms, typically held in optical traps, as qubits for quantum computing applications.
6. Photons: The focus of this article is on the photonic platform. Photonic quantum bits (qubits) are flying qubits, and it is advantageous to measure them quickly after creation due to their high mobility. Unlike stationary qubits in other platforms, photons travel at the speed of light, necessitating a measurement-based quantum computing model to suitably harness their properties.

Photonic Quantum Computing — Two Approaches

Continuous variable photonic quantum computing offers two distinct approaches for qubit encoding:

1. DV (Discrete Variable) Encoding: In this approach, information is encoded into two levels of the system. The photonic mode is represented by a harmonic oscillator, which has multiple energy levels. However, only two of these levels are populated to create a two-level encoding. Alternatively, qubits can be encoded using the polarization of single photons. A drawback of this approach is the need for a large number of physical qubits to create a logical qubit, making error correction challenging.
2. CV (Continuous Variable) Encoding: The CV encoding approach utilizes photonic modes represented by harmonic oscillators with multiple energy levels. Multi-photon states are employed to access many levels of the harmonic oscillator, creating multi-level encoding. A notable advantage of this approach is the ability to perform error correction in a single mode, avoiding the requirement for a large number of physical qubits. However, producing the GKP state, a specific multi-photon state used in CV encoding, can be challenging.

Computation via Teleportation — The Fundamental Ingredient

Measurement-based quantum computing relies on the concept of state teleportation, which serves as the cornerstone for both DV and CV photonic quantum computing paradigms. The process begins with the preparation of an entangled state known as the EPR state, and in the context of continuous variables, it is referred to as the two-mode squeezed state. This entangled state serves as the quantum resource for the teleportation process. The teleportation procedure involves utilizing half of the entangled state and subjecting it to a Bell state measurement. The other half of the EPR state, representing the input qubit, interacts with the measured output through a beam splitter.

Teleportation Circuit: Image Credits [1]

Homodyne measurements are performed on the beam splitter outputs, which correspond to the continuous variable quadrature, namely the phase and amplitude quadrature of the state. The specific quadrature being measured depends on the phase setting of the homodyne detector, and multiple phase settings can be employed with two homodyne detectors. During the teleportation process, a unitary transformation is applied to the input state, induced by the measurement scheme of the two homodyne detectors. This unitary transformation comprises a combination of rotation gates, squeezing gates, and another rotation gate, parametrized by angles theta plus and theta minus, which are directly determined by the measurement settings of the homodyne detectors.

By manipulating the measurement/phase settings of the detectors, distinct unitary operations are introduced to the teleported state, effectively enabling the implementation of single mode/single qubit gates in a measurement basis. For instance, setting the two angles to zero results in an identity transformation, merely teleporting the input state without alteration. However, deviating from these standard settings introduces the single mode operation.

The EPR State Generator: Squeezed Light Sources

The EPR state is produced using two squeezed light sources that combine at a beam splitter, generating entanglement. Squeezed states can be illustrated in a phase space diagram, similar to a phase space diagram for mechanical oscillators, where x and p represent the amplitude and phase quadrature for photonic states. Classical states are characterized by definite x and p values, while quantum states entail uncertainties and quantum fluctuations, governed by the Heisenberg uncertainty relation.

For squeezed states, the uncertainty region is “squeezed,” resulting in reduced uncertainty in one variable (quadrature) and increased uncertainty in the conjugate variable, while still adhering to the Heisenberg uncertainty principle. The amount of squeezing is crucial for the fidelity of implementing single mode gates through unitary transformations, with greater squeezing needed to attain noise-free transformations. Although infinite squeezing is theoretically ideal for noise-free transformations, practical limitations impose a maximum squeezing threshold, inevitably introducing some noise that must be mitigated through error correction methods.

EPR State Generator: Image Credits [1]

Brief Overview of the Detection System: Homodyne Detector

The homodyne detector comprises two detectors and 50–50 beam splitters. The input signal meets the local oscillator, enabling control over the local oscillator’s phase. By manipulating the local oscillator’s phase, different quadrature can be measured. These quadratures may include the amplitude quadrature (squeezing measurement) or the phase quadrature (anti-squeezing measurement), with the ability to measure any other quadrature in between. The choice of quadrature measurement depends on the specific requirements of the teleportation process and gate implementation.

Balanced homodyne detection process is as follows: A local oscillator (LO) field with state |αω 0 and controlled phase ϕ is combined with the quantum field modes of interest using a 50/50 beam splitter. The information about the quantum field is obtained by subtracting the photocurrents after this combination. Image Credits [2]

Graphical Representation of EPR and Cluster States

In photonic quantum computing, the EPR state is visually represented by two connected dots, while the input state is depicted as a green dot, coupled to the EPR state through a beam splitter. The teleportation circuit is graphically illustrated, enabling the implementation of unitary transformations. By concatenating the circuit, multiple single-mode gates can be implemented in a sequence of teleportation circuits, forming a long chain of entangled nodes known as the 1D cluster state. These cluster states play a vital role in measurement-based quantum computation.

Measurement-Based Quantum Computation

The 1D cluster state comprises a series of entangled nodes connected by long links, creating a resource for implementing unitary transformations through homodyne measurements along the state. Sequential teleportation steps are performed throughout the 1D cluster state, facilitating single-qubit operations. However, for universal quantum computing, two-qubit operations are required, necessitating a 2D cluster state. By coupling an input register of qubits to the 2D cluster state, homodyne measurements of all nodes allow for both single-qubit and two-qubit operations, forming the basis of measurement-based quantum computing. The concept of measurement-based quantum computing was introduced over two decades ago.

Image Credits [1]

Cluster State Generation

In laboratory settings, generating a 2D cluster state requires careful consideration. A single link in a cluster state can be achieved using two squeezed beams on a beam splitter. To create a square or larger cluster state, four squeezed beams are needed to interfere in an interferometer linear network. Generating a cluster state with N nodes necessitates N squeezers and N homodyne detectors. However, scaling up to a large quantum computer, e.g., with a million qubits, using free-space bulk optics or photonic circuits may encounter limitations. To overcome this, various multiplexing techniques need to be employed.

a) The researchers implemented a one-input Multiplexed 2-Dimensional Cluster State (MBQC) using a 1-dimensional cluster state. b) They also demonstrated a universal multi-input MBQC using a 2-dimensional cluster state. In the diagrams, each colored circle represents a mode, while each link indicates quantum entanglement. Image Credits [3]

2D Cluster State Generation

Cluster states are a crucial resource in quantum computing, particularly for one-way quantum computation. The generation of 2D cluster states using photonic quantum computing involves the creation of squeezed states of light followed by a series of interference and delay operations. The process is formally explained as follows:

1. Squeezed State Generation:
The first step is to produce squeezed states of light. This is achieved using a parametric down-conversion process inside high-quality cavities, also known as optical parametric amplifiers or optical parametric oscillators. The parametric down-conversion process generates squeezed states with reduced uncertainty in one of the quadratures of the electromagnetic field, below the shot-noise limit. Two independent squeezed state generators are used, each producing a squeezed state of light.

2. Interference on Beam Splitters:
The two squeezed modes from the generators are then combined using a beam splitter. The interference on the beam splitter results in the creation of two-mode squeezing or Einstein-Podolsky-Rosen (EPR) states. Graphically, the squeezed states are represented as single dots produced in time, where the distance between the dots corresponds to one clock cycle (repetition rate of the laser system).

3. Formation of 1D Cluster State:
By connecting two squeezed states using the beam splitter, the EPR states are generated. These EPR states serve as the building blocks for the cluster state. Next, one of the modes is delayed with respect to the other mode using a long piece of fiber. The length of the lower fiber is set to be one clock cycle longer than the other one. The delayed modes are then made to interfere again on a second beam splitter (BS2), creating new correlations. This setup results in the formation of a 1D cluster state.

4. Creation of 2D Cluster State:
To generate a 2D cluster state, an additional delay line is introduced, which is much longer than the first delay line (n times the clock cycle). The two 1D cluster states are then made to interfere again. To visualize this process, the 1D cluster states are rolled onto a cylinder. The diameter of the cylinder is determined by the delay introduced in the second interferometer, with N nodes on the signal variants of the cylindrical graph state. This forms the 2D cluster state.

The scheme for 2D cluster state generation involves the following steps: 1. Produce squeezing using OPOA and OPOB, coupled into a fiber (97% efficiency). 2. Interfere temporal modes in spatial modes A and B with fiber-coupled beam splitters (generate EPR-states at BS1). 3. Entangle EPR pairs to form a 1D cluster state with τ delay in mode B and BS2.4. Transform the 1D cluster state into a 2D cluster state with Nτ delay and BS3. 5. Measure temporal mode quadratures using homodyne detectors (HDA and HDB) for nullifiers calculation. 6. Experimental setup: Short 50.5 m fiber (247 ns duration) for initial squeezing, long 606 m fiber (N=12) for 2D cluster state. 7. Temporal modes defined by asymmetric function (less than 10^(-3) mode overlap). Image Credits [4]

5. Operations on the 2D Cluster State:
The resulting 2D cluster state allows for single-mode and two-mode operations through measurements. Homodyne detectors are employed to measure and perform computations on the cluster state surface. The 2D cluster state serves as a resource for one-way quantum computation, enabling the implementation of quantum algorithms and information processing tasks.

Implementing Gates

The implementation of gates involves manipulating the cluster state to perform single-mode computation and single-qubit operations.

1.Carving Out Lines in the Cluster State:
To perform single-qubit operations, lines are carved out in the cylindrical cluster state. This is achieved by measuring and decorrelating specific nodes from the rest of the state. By measuring these nodes with a specific phase setting and local oscillator phasing, correlations to certain nodes in the cluster state are removed, effectively carving out lines in the state. Along these lines, single-qubit operations can be performed.

a) The QPU utilizes a quantum circuit decomposed into gates applied to a cluster state using projective measurements of input and cluster state modes. b) A 1D coiled-up cluster state is generated at the logic level to encode quantum information. Computation occurs using a two-mode measurement device with a beam-splitter (BS3) and two homodyne detectors measuring in bases θA,k and θB,k (where k represents the temporal mode number). Additional experimental setup details can be found in SI section 1. c) The coiled-up cluster state in the logic level allows encoding of input states, |ψ_0⟩, . . . , |ψ_5⟩, on the circumference. Control modes are measured to project the cluster state into wires, enabling the implementation of single- and two-mode gates through gate teleportation. Spatial modes A and B are represented by bright and dark nodes, respectively, with red arrows indicating the BS3-operation of the measurement device. d) and e) Cut-outs of the cluster state demonstrate the implementation of single- and two-mode gate operations, Uˆ and Vˆ, based on the measurement device’s basis settings. The coiled-up cluster state at (1) is projected into wires at (2) by measuring odd temporal control modes in the θc control basis before the gate implementation at (3). Image Credits [5]

2. Teleportation and Unitary Transformations:
Once the lines are formed, teleportation is employed to perform unitary transformations on the input state. The input state is coupled into the two-mode squeezed state along the lines, enabling the implementation of a unitary transformation through teleportation. This process can be repeated as long as the cluster state is being produced. The teleportation circuit is represented by the links along which processing is performed.

3. Gate Set and Symplectic Matrix:
Different unitary transformations corresponding to qubit gates are demonstrated in the experiment. The rotation gate, shear gate, and squeezing gate are shown to form a universal gate set for single-mode operations. These gates are characterized by a symplectic matrix consisting of four entries. The experimental measurements of these entries are compared to theoretical predictions, showing a strong agreement between experiment and theory.

In this figure, the researchers investigated single-mode gates. They used symplectic matrices to characterize the implemented rotation, shear, and squeezing gates, measured through gate tomography (as described in SI section 3). The gate noise variance in each output quadrature for each gate was measured and compensated for by -6 dB to account for the gate noise matrix. The measured gate noise was compared to the initial momentum squeezing variance, and the results were obtained from 10,000 measurements. The single-mode gate noise was also studied as a function of pump power, showing the effect of squeezing and vacuum conditions. Green and purple lines represented expected gate noise for more optimal parameters and different squeezing bandwidths. Image Credits [5]

4. Two-Mode Operations:
Two additional gates are implemented using a parallel teleportation circuit. By combining two parallel teleportation circuits with a beam splitter operation, two-mode operations can be achieved. This involves carving out a different graph structure within the cluster state to implement two-mode operations on the input states.

5. Circuit Setup:
With the ability to implement single-mode and two-mode operations, circuits with multiple modes can be set up. By performing two-mode gates between different inputs, various circuit configurations can be realized. The flexibility of setting up different circuits is achieved by adjusting the phases of the local oscillator in the homodyne detectors. The cluster state remains unchanged throughout the circuit, with only the phases being modified.

A quantum circuit was designed to encode a logic qubit using the 3-qubit bit-flip error correction code, with input qubits encoded in the GKP (Gottesman-Kitaev-Preskill) scheme. The circuit was rewritten in terms of continuous-variable gates, namely FˆFˆCˆZ and Fˆ±1 = Rˆ(±π/2). The encoding circuit was implemented on three coupled cluster state wires, and circuit tomography was performed to estimate the resulting circuit symplectic matrix using quadrature correlations of input and reference modes. The measured gate noise was compared to the expected gate noise for a vacuum state instead of the cluster state. Additionally, the gate noise was compensated for by the combined circuit gate noise matrix N, represented by purple bars. The compensated gate noise was also compared with the initial squeezing variance for cluster state generation, which was -4.4 dB (dashed gray line). Image Credits [5]

6. Noise in Gate Implementation:
A challenge in gate implementation arises due to the finite squeezing in the EPR state generator. When the squeezing is not infinite, noise is introduced during computation. The graph illustrates that gate noise increases as a function of the strength of the two-mode gate. Overcoming this excess noise becomes necessary for effective quantum computation.

Noise Addition and Error Correction

Excess noise can be introduced during quantum operations, which can affect the accuracy and reliability of computations. To address this issue, error correction techniques are employed. Consider a squeezed state, where one quadrature (x-direction) is squeezed, and the other quadrature (p-direction) is anti-squeezed. When this squeezed state is sent through a processor, the processor’s operations introduce noise, resulting in a noisy output state. To restore the original state and correct for the added noise, a quantum non-demolition operation is utilized.

Noise addition and Error Correction. Image Credits [1]

The quantum non-demolition operation allows for non-destructive measurement of the amplitude quadrature (X direction). Once this measurement is performed, corrections can be made to account for the added noise in the system. However, for this error correction scheme to work effectively, a squeezed resource must be added. Without the addition of a squeezed vacuum in the empty port of the beam splitter (used in the quantum non-demolition device), vacuum noise would be introduced, and the correction for external noise would not be possible. By including the squeezed vacuum in the empty port, it becomes feasible to correct for lost squeezing and regain the original squeezed state, thereby correcting the output state.

However, this error correction process introduces a challenge in that while eliminating errors in the X quadrature, it adds extra noise to the P quadrature. Since quantum information is encoded in both X and P quadratures, it becomes crucial to be able to correct errors in both quadratures simultaneously. To achieve this, the GKP state is considered, as it is somewhat squeezed in both directions.

GKP Qubits

The GKP state is a superposition of many squeezed states, with each state in the superposition representing a logical eigenstate (0 or 1) of the qubit. While each logical eigenstate has a single peak representing a squeezed state, the GKP state combines multiple peaks, resulting in a coherent superposition of squeezed states in both X and P quadratures. This feature is illustrated in the phase space diagram for the GKP state, where every dot in the grid-like structure represents a squeezed state in both quadratures.

Image Credits [1]

Due to the simultaneous squeezing in both X and P quadratures, GKP states become powerful resources for quantum error correction. They allow for the correction of noise or errors that affect conjugate quadratures (X and P). Consequently, the production and utilization of GKP states as qubits in the quantum computing scheme are highly desirable, as they offer robustness against noise and errors. The generalized qubit is then defined as a superposition of two logical eigenstates, which is the fundamental unit of quantum information processing.

Bosonic Error Correction

Error correction plays a pivotal role in the realm of photonic quantum computing, particularly when handling states like the Gottesman-Kitaev-Preskill (GKP) state. The process of error correction revolves around mitigating the noise that arises during computation, leading to the smearing of individual peaks (squeezed peaks) within the GKP state. To address this challenge, the application of GKP error correction becomes essential.

Image Credits [1]

The GKP error correction circuit distinguishes itself from the teleportation circuit by utilizing a unique entangled state known as the census data. Unlike traditional EPR states entangled with the squeezed state, these phi-not states serve as the resource states in the teleportation circuit. When these states interfere in a beam splitter, a novel entangled state, the GKP entangled state, emerges, which serves as the resource for the teleportation-based error correction circuit.

The implementation of GKP quadrature correction is achieved by qubit teleportation. Qunaught states, |∅⟩_GKP, prepare a two-mode GKP-qubit Bell state. After the nm-delay in the computational level, the GKP-qubit state to be corrected, |ψ⟩_GKP, overlaps in time with part of the Bell state. Using a Bell measurement with the TDMD, |ψ⟩_GKP is teleported through the Bell state, resulting in a purified GKP qubit state based on Eq. (6). This process corresponds to a graph in the 3D time lattice and a corresponding circuit diagram. Image Credits [6]

The error correction process commences with the transformation of the noisy state into a cleaner version. Measurement of the noise in the two homodyne detectors follows, and subsequently, a feed-forward loop is employed to correct for the noise. Remarkably, this error correction process is seamlessly integrated within the cluster state, showcasing the effectiveness of the measurement-based approach for error correction.

During the error correction process, the logical one state may inadvertently emerge alongside the desired logical zero state. To rectify this, standard qubit error correction schemes are utilized, complementing the error correction for gate noise affecting the teleportation circuit. As a result, two error correction schemes work in tandem to effectively eliminate errors from the state.

Architecture and Fault-Tolerant Threshold

The proposed architecture for generating the cluster state takes into account both error correction and computation in photonic quantum computing. To enable universal computation and address continuous error correction as well as qubit noise error correction, a 3D cluster state structure is necessary. A 2D cluster state proves inadequate for this purpose, requiring an additional dimension to accommodate qubit error correction. This is achieved by employing surface encoding to encode logical qubits on one surface of the 3D cluster state.

The scheme consists of three parts: resource preparation gadget, computational level, and temporally delocalized measurement device (TDMD) for gate implementation. The setup uses temporal multiplexing of two spatial modes, A and B. Wires with two-mode entanglement are shown in the time domain, forming a 3D time lattice for computation and gate teleportation using the TDMD. Image Credits [6]

A critical parameter for quantum computing is the fault-tolerant threshold, which has been calculated for this system with respect to the squeezing variance of the cluster state. Adequate squeezing is crucial for fault-tolerant computation, as insufficient squeezing introduces excessive noise. The fault-tolerant threshold for quantum computing with this architecture is determined to be 12.7 dB squeezing. Consequently, suppressing the squeezed quadrature by 12.7 dB or more becomes imperative to enable fault-tolerant quantum computing using this setup.

The simulated logic Zˆ and Xˆ error probabilities of the surface-4-GKP code are plotted against the squeezing level of the |0⟩ sq -states used for gate implementation, the GKP qubits encoding the surface code, and the |∅⟩_GKP-states used for quadrature correction. The logic error probability is shown for various code distances (d), and the fault-tolerant threshold, where the logic error rate decreases with increasing code distance, is observed at 12.7 dB of squeezing. Error bars representing standard deviations are estimated using bootstrapping. Image Credits [6]

Challange: Generation of error-correctable state — GKP state

Implementing this photonic quantum computing scheme presents a significant challenge in generating the GKP state. While constructing the entire circuit, encompassing computation and error correction, is relatively straightforward with linear optics and homodyne detectors, generating the GKP state proves to be the primary obstacle. Several methods exist for generating the GKP state, one of which is the Gaussian Boson sampling technology.

One way to generate GKP state Image Credits [1]

Gaussian Boson sampling technology entails generating squeezed input states by squeezing the vacuum state, followed by their interference in a linear network of beam splitters and linear optics (utilizing multiple input states). Subsequently, all output modes are measured, except one. The waiting period involves observing a specific photon number pattern, indicating the successful projection of the last mode into a GKP state. Nevertheless, this approach is probabilistic, with a success rate as low as 1% or even lower, necessitating multiplexing to enhance the likelihood of deterministic generation of GKP states.

Pros of Photonic Quantum Computing

Photonics in quantum computing offers significant advantages, particularly when employing CV Bosonic encoding. These advantages are rooted in its unique properties, leading to enhanced performance in various aspects of quantum information processing.

1. Low overheads for fault-tolerance: One key advantage of photonics-based quantum computing using CV Bosonic encoding is the reduced requirement for qubits to implement error correction. This is due to the inherent resilience of photons against decoherence and noise, minimizing the number of qubits needed for robust fault-tolerant operations.

2. Room temperature computation: Photonics-based quantum information processing can be conducted at room temperature. While generating the actual qubits may require cryostats and low-temperature setups offline, the Quantum Processing Unit (QPU) itself can function using equipment at room temperature. This offers practicality and simplifies the operational environment compared to other quantum computing approaches that necessitate extremely low temperatures for the entire computation.

3. Modularity and networkability: Photons possess the exceptional property of easy transmission through optical fibers. As a result, different modules in the quantum computing system can be readily connected, allowing for seamless integration into larger networks. This modularity and networkability provide flexibility in building scalable quantum computing architectures.

4. Scalability: Photonics-based quantum computing demonstrates excellent scalability potential due to its connectable nature. Various multiplexing techniques, such as employing different frequency modes, spatial modes, and temporal modes, enable efficient integration of multiple quantum elements. This scalability capability is crucial for advancing quantum information processing towards more complex and powerful computations.

Conclusion

In conclusion, photonic quantum computing, specifically in the context of continuous variable (CV) encoding, shows promising potential due to its advantages over other quantum computing platforms. CV quantum computing leverages the unique properties of photonic modes, such as the speed of light and easy networkability through optical fibers. The implementation of photonic quantum computing relies on the concept of cluster states, which serve as the resource for computation and error correction. The key challenge lies in the generation of error-correctable states, such as the GKP state. The current state-of-the-art approach is Gaussian Boson sampling, which offers a probabilistic method for generating GKP states. However, for practical quantum computing, deterministic and scalable methods of GKP state generation are required. Overcoming this challenge will be crucial for harnessing the full potential of photonic quantum computing, paving the way for fault-tolerant and scalable quantum information processing at room temperature.

References

Photo by Sigmund on Unsplash

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