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Noisy intermediate scale quantum (NISQ) systems are susceptible to errors that culminate in near-one hundred percent data loss. This is due to quantum state fragility and the incredibly high quantum communication error rates caused by decoherence, or quantum noise. As such, stabilizing qubit operational imprecision in quantum information processing is a critical area of research in quantum computing. Adaptive quantum machine learning (QML) methods, like unsupervised and fully entangled quantum generative adversarial networks is one such technology theorized to provide a breakthrough in quantum error suppression. Mechanizing the quantum error detection and correction process with QML provides a path forward from today’s monolithic quantum computers running almost exclusively single-core quantum processing unit (QPU) designs, to the next generation of federated quantum computers using multi-core QPUs. Automating the detection and correction of quantum errors in powerful NISQ devices will pave the way for fault-tolerant quantum computing, making quantum speeds at quantum scale suddenly achievable.
*Address all correspondence to: theresa.melvin@toriitechnology.com
1. Introduction
The mere mention of quantum speeds at quantum scale when referencing today’s noisy intermediate scale quantum (NISQ) devices, which are susceptible to environmental errors so severe that they result in near-one hundred percent data loss will grind a room full of quantum researchers to a steadfast halt. Google’s AI Quantum team found this out when they asserted “Quantum Supremacy” [1] in 2019, basing their Sycamore fidelity results on 1% signal in 99% noise. Google’s work, however, provided the quantum research community with valuable insight. Most notably, it forced quantum developers to stop and ask: what is it going to take to achieve a quantum computer that is not only as reliable and as scalable as an ordinary classical computer, but as fast as a quantum computer?
This chapter explores quantum computing technology, the errors that inhibit this technology, and the adaptive quantum machine learning (QML) methods developed as an answer to stabilize this qubit operational imprecision. Special attention is paid to unsupervised and fully entangled quantum generative adversarial networks, and how they have emerged over the past 5 years to become an important QML technology in quantum error correction.
This chapter is organized as follows: section two begins with an overview of the quantum information process, followed by section three’s quantum errors on NISQ devices. Section four focuses on quantum error correction research, a necessary background for section five’s quantum machine learning discussion. This then leads to the unsupervised quantum generative adversarial network noise (uQGAN) model experiment in section six, followed by chapter summary in section seven’s conclusion.
Quantum bits or qubits are fragile and packaged with quantum information (QI). Prior to transmitting this QI through a qubit communication channel, the QI is encoded. Next, quantum error detection and correction (QED/C) methods, like stabilizer codes, continuously measure the quantum waves, called syndromes [2]. While the quantum states are not directly measured since that would cause the amplitudinal wave to collapse into a particle [3], the parity of qubits is automatically adjusted based on measurements to correct the quantum state [4]. The QI is then decoded, and the transmission is complete. Thus, stabilizer code performs a rudimentary form of QED/C which successfully suppresses quantum errors (QE/s) during quantum information processing (QIP) execution against trivial programs. This solution, however, lacks scale due to the high number of physical qubits [5] required to encode a single logical qubit.
Presently, the qubit communications channel used in QIP consists of a hard-wired and direct integration into a single-core quantum processing unit (QPU) [6]. Due to its convoluted integrated network design, this single-core QPU design is rife with challenges and requires manual intervention for quantum interactions [7]. This configuration generates noise, vibrations, and temperature fluctuations that generate errors, faults, and decoherence [8] causing irreparable damage to the QIP operation, stifling QC performance.
A quantum communication error rate (QCER) benchmark is used to observe the QIP operation, and it measures the probability that something went wrong with the qubit during the quantum gate operation (I.e., quantum error rate) against how accurately the actual output matches the desired output (I.e., quantum state fidelity). Unlike classical calculations, which are deterministic (predictable) and produce expected results and outcomes, QC computations are non-deterministic (random) and therefore do not possess any specific outcome. As such, QCERs are exceptionally high during QIP due to the inherently brittle state of qubits, illustrating the stark immaturity of modern QC technology. To advance quantum computing to practical use, efficient QED/C methods are needed for common QEs to reduce QCERs and safely teleport QI during the QIP.
This section explores the types of quantum errors observed in superconducting phase qubits, one of the most popular quantum computing technologies in use. Errors represent an organic part of any computer operation. However, QC errors are considerably more complex than classical errors. Disturbances that cause QC errors occur when quantum states are not prepared properly, they exceed their target, or they drift away [9]. For this reason, the QEs are often distinct to the underlying QC hardware, and any associated error detection and correction methods would therefore need to be equally subjective.
3.1 Qubit-based QC technologies
The largest QCs currently in use are gate-based quantum computers, also known as near-term QCs, or noisy intermediate-scale quantum (NISQ) devices [10, 11]. These systems are built with superconducting qubits, trapped ions, spin qubits, semiconductors, photonics, nitrogen-vacancy center (NVC), nuclear magnetic resonance (NMR), and quantum dots [4, 12, 13], to name a few.
Current NISQ systems possess anywhere from dozens of qubits to 40 qubits like one core of Rigetti’s dual-core QPU [14, 15], to 54 qubits, like Google’s Sycamore [16], to 433 qubits, like IBM’s Osprey, the largest QC in existence [17] as of the time of this writing. Yet, even with 80 qubit dual-core QPUs [15] and 4000 qubit QC systems planned [17] it should be noted that NISQ devices will still not be advanced enough to achieve fault tolerance without a significant reduction in QEs. This will preclude the level of scale needed for quantum supremacy [1], where problems that elude classical computers are consistently solved in a timely manner. Likewise, with near-total data loss, it remains equally challenging to achieve any kind of quantum advantage [18] in a commercial setting until the QE issue is suitably addressed.
3.2 Quantum error correction for NISQ devices
The focus of this chapter is leveraging QML for QEC in superconducting phase qubits, one of the most popular NISQ QC technologies in use today. The three primary methods used to address QEs on this type of NISQ device are, error:
Suppression
Mitigation
Correction
This chapter focuses exclusively on the third type of QE: quantum error correction (QEC) since it is closely associated with the field of QML. Quantum machine learning intersects the fields of quantum computing with ML. For this reason, QML is posited as one possible method of detecting quantum errors as they occur and then summarily correcting them in real-time. The challenge with putting this theory into practice is the stark number of NISQ QEs and the immense complexity behind all these errors. A glimpse into the difficulty of this developmental task is highlighted below.
3.2.1 Common causes of decoherence on NISQ devices
Quantum processors on NISQ devices generate noise, vibrations, and temperature fluctuations [7, 18]. This in turn creates errors, faults, and other types of communications failures, leading to exorbitantly high QCERs, which stifles QC performance [19]. As a result, very few non-trivial quantum programs are executed to completion. Moreover, non-trivial programs that are executed to completion rarely obtain the correct results. This is due to environmental noise, also called decoherence, which creates an assortment of QEs that interfere with the NISQ device. A few of the most common errors are discussed in turn below.
The three most common types of quantum errors are bit-flip errors (bfe/s), phase-flip errors (pfe/s), or a combination of the two, known as bit-and-phase-flip errors (bpfe/s). While both classical and quantum computers contend with bfes [4] which modify the original binary state from a zero to a one or a one to a zero, QCs must also contend with pfes, which modify the quantum state, changing the qubit from a positive to a negative, (or vice-versa). Phase-flip errors in turn represent the most prevalent type of QC error, and as such most QED/C research focuses on pfe prevention [20]. However, the third and most difficult type of quantum error to detect and correct is the combined bpfe, in which both the original and quantum states are altered during QIP [3]. While combination errors are rare, they represent one of the most difficult QED/C problems to solve.
In addition to the above QEs, many other errors impact QCs. These include, but are not limited to gates, mixed unitary, phase damping, amplitude damping, depolarization, asymmetric depolarization, and resets [2, 20]. This extended error list stems from QM, with properties like superposition that allow the quantum state to be many different states simultaneously. The relative phase and amplitude of the superimposed states ultimately determines the properties of the entire quantum state. These QC devices can generate error in many ways, for many different reasons, with different Gates (Hadamard, Pauli, R, etc.) all providing their own individual errors.
Next, according to [7] phase damping errors occur when the coherence of the quantum state is destroyed due to the relative phases of the superposed states randomly changing with time. In turn, amplitude damping errors occur when the quantum state changes due to environmental energy dissipation. Depolarization errors, or the sudden death of maximally entangled qubits [21] results in the loss of QI. Furthermore, this noisy channel can be either one-sided (symmetric) or multi-sided (asymmetric) [8]. For these types of NISQ specific device errors, the classical error correction techniques are ineffective at correcting quantum errors.
Quantum error detection and correction techniques are required to protect qubits from decoherence and noise during QIP, facilitating the successful teleportation of QI [3, 13]. Unfortunately, many obstacles exist to making QCs error-resistant or fault-tolerant. First, the no-cloning theorem [22] in QM prohibits an unknown quantum state from being reliably copied without destroying the original quantum state. Therefore, duplicating an arbitrary unknown quantum state is forbidden, meaning quantum data cannot be copied and pasted between qubits the way classical data is replicated using classical bits. Next, the Heisenberg uncertainty principle dictates that unknown quantum states cannot be fully measured without inducing a QE and destroying the state [2, 20, 23]. As such, unlike a classical experiment, it is impossible to observe a quantum experiment without destroying the experiment.
Quantum error correction is a research area addressing real-time quantum state feedback and in-sequence measurements necessary for QED/C. While there are many different types of error correction for classical systems, the most common fault-tolerant approaches are checkpointing, parity checks, error correction code (ECC), and redundancy [4, 5, 8, 18, 20]. Checkpointing cannot be used on QCs for either QED or QEC due to the no-cloning theorem. In turn, parity checks require an extra (parity) bit to be attached to the data for redundancy. A count of each data’s parity bit confirms if the data was successfully transmitted. Parity checks do work for detecting errors on QC; however, parity checks do not offer a method for correcting quantum errors. To determine the parity count of the QI requires a measurement. This violates Heisenberg’s uncertainty principle, which asserts that an unknown quantum state cannot be measured without destroying the state.
Next, ECC encodes the entire byte stream comprising the data transmission into a single word. ECC does work for QCs, but only for bit flips, where the qubit changes from a 0 to a 1 (and vice-versa). ECC uses a majority voting system, flipping the corrupt (non-matching) bit to match the other (matching) bits. ECC, however, does not work for quantum phase errors, where the qubit changes from a positive to a negative (and vice-versa), since phase errors do not exist in classical systems.
The final classical error correction technique discussed is redundancy. Redundancy simply sends the same data repeatedly to ensure transmission success. While this method does work for QC at a small scale, satisfying both QED and QEC, redundancy is impractical due to the massive number of physical qubits required to support the repeatability of QI transmissions. To achieve quantum scale would require several orders of magnitude more physical qubits due to interactions among nonadjacent qubits operating on and correcting redundant encoded quantum data. This level of massive redundancy is cost-prohibitive for QCs.
4.1 Stabilizer codes
Stabilizer codes are a quantum error correction encoding process affording QI redundancy for a single physical qubit. The stabilizer code wraps the physical qubit containing the QI with multiple logical qubits. Using stabilizer code, the state of a single physical qubit is encoded with several ancilla or helper qubits, creating a single stabilizer code for quantum error correcting code (QECC). The number of ancilla is determined by the stabilizer code used, with four, seven, and nine qubit models common, depending on the model selected and the type of quantum error to be corrected. For instance, the five-qubit stabilizer code is the smallest model capable of correcting either bit-flip or phase-flip errors, but only for a single quantum error. This is distinguished from Shor’s original nine-qubit code, which corrected for two-qubit errors.
4.2 Surface codes
Topological codes, such as surface codes, overcome the quantum-scale challenges of stabilizer codes. Surface codes leverage a two-dimensional lattice of qubits with nearest-neighbor coupling, affording integrated QECC. Due to the integrated qubit design, surface codes scale well, revealing a high tolerance for locally correctable quantum errors. However, surface codes are in their infancy and are expensive to produce. This renders the technology largely out of reach for the mainstream quantum research community.
4.3 Quantum machine learning models
Neural decoding algorithms and generative adversarial networks (GAN/s) are two types of ML models with demonstrated, albeit small-scale, QECC success [11, 13, 24]. The QML models are trained on quantum error-injected data, the quantum algorithms eventually learn the probability distribution of the QEs and correct the error chains to recover the correct quantum states. In theory, this is one way to produce a fault tolerant NISQ system. This is the underlying premise of the section that follows.
Traditional ML merged mathematical algorithms with statistical modeling, providing a method for computers to learn complex tasks with minimal instruction. Quantum machine learning is an emerging theoretical field combining quantum algorithms with traditional ML. Just as ML techniques are used on classical computers for big data processing, QML is envisaged to leverage qubits and multi-core QPUs to support the computational needs of next generation full-stack quantum computing workloads. However, QML’s necessary quantum algorithm (QALGO) innovation has remained slow to advance due to the lack of realistic QC technology and decoherence struggles. These limitations impede QML execution, performance, and evaluation of complex real-world algorithms.
With advancements in QML and QC research on the rise, practical quantum uses for industry are starting to emerge, prompting further investment in the field. As more companies discover quantum applications capable of revenue generation, quantum will eventually be adopted into mainstream industry. This increased QC demand for commercial access will drive further QC innovation, rapidly advancing QALGO development and moving QCs closer to achieving a total quantum advantage over all classical systems.
5.1 Quantum algorithms
Quantum algorithm development has been an active area of research since 1994, when Shor’s Algorithm exponentially sped up integer factor detection for a large number [25]. With quantum computing still in its infancy, hundreds of QALGOs exist today for many different use cases. As such, anything a classical computer can do today, a QC can also do [26]. Likewise, provable quantum advantage is known for a few dozen QALGOs [18] though innovation remains slow to advance due to the lack of realistic QCs, which impedes real-world QALGO performance and evaluation. While quantum developers work to bridge this QC hardware gap using classical computers, classical bits lack the quantum mechanical properties of quantum bits, which renders their hybrid quantum-classical algorithm (HALGO) work deficient and marginally irrelevant to QALGO development.
Unlike HALGOs, QALGOs completely rely on unnatural quantum physics principles, which are counterintuitive to classical physics. The QALGO’s non-intuitive and strange behavior facilitates quantum’s massive scale [18, 27, 28]. In turn, HALGOs remain computationally limited by their classical computing, which is constrained by the laws of contemporary physics. For instance, QC-generated probability distributions are hard for classical computers to sample since comparable quantum mechanical conditions do not exist in a classical computer system.
Next, probability distributions for a classical ML problem possess either a positive or negative. In turn, QML probability distributions leverage the QM property of superposition, allowing QALGO results to be positive, negative, or a combination of the two (superposed). HALGOs must approximate the quantum probability distribution since the classical hardware is physically incapable of simulating a large number of quantum wave excitations [29]. This is problematic since the accuracy of these estimates remains largely unknown and borders on conjecture. While QALGO development remains challenged by the lack of mature QC hardware capabilities, QALGO studies continually produce innovative potential. One such area of QALGO advancement is recent quantum generative adversarial network (QGAN) research.
5.2 Quantum generative adversarial networks
A generative adversarial network (GAN) is a type of semi-supervised or unsupervised neural network used for creating high-quality synthetic data. Derived from a novel zero-sum game concept, a single GAN framework is comprised of two independent neural networks, called a generator (G) and discriminator (D). Each GAN neural network (the G and the D) then attempts to outperform the other, forcing each neutral network to continually retrain. This process repeats until the GAN model achieves no further performance gain.
Generative adversarial network technology has matured considerably over the past decade, with GANs now successfully modeling molecular synthesis [30] and dark matter [31]. They are further used to: detect and assess cyber-risk [32], identify counterfeiting and fraud [33], accelerate financial time-series modeling [34], and explore human-centered computing [27]. Likewise, GAN technology has shown success with photorealistic image creation and enhancement [35], video prediction [27], and audio processing [22]. GAN application has even successfully branched into natural language processing (NLP) and natural language understanding (NLU) [21], where it summarizes text and generates semantics. While GANs have become a prolific data creation [36] tool in AI, their application remains limited due to both GAN complexity and classical system performance limitations.
Challenges surrounding GANs include vanishing gradients, mode collapse, and failures to converge or reach a Nash Equilibrium [28, 31, 37]. Vanishing gradients are common when the G fails to train due to an ineffective D [31, 38]. When a G lacks heterogeneity among its generated samples and supplies the D with only a single sample category (mode), either a partial or complete mode collapse occurs. The difference between partial and complete mode collapse is that some (partial) or all (complete) of the G’s generated samples are mapped to identical D-output. Lastly, GANs routinely fail to converge, oscillating from one sample generation to the next without achieving any type of equilibrium.
To date, most QGAN research has been HALGO rather than QALGO-focused [31, 38]. Because HALGOs are leveraged, a hybrid quantum-classical GAN (HGAN) is created. These HGANs are neither entirely classical nor entirely quantum since a portion of the GAN process is executed on a classical computer. At the same time, the remainder of the GAN execution occurs on a QC. As a result, many of the same challenges that plague a classical GAN (CGAN) also encumber HGANs. Moreover, as discussed below, HGANs have the additional and arduous QML challenge of encoding classical data into a quantum state.
For these reasons, the goal has been a true quantum GAN, with both the G and D running on a parameterized or variational quantum circuit (VQC). Recent QLAGO research on entangled QGAN models has been groundbreaking [31, 38], successfully circumventing common GAN errors. Based on recent QGAN innovations, a QML-mechanized solution for QED/C, placing equal emphasis on QALGO and NISQ hardware, seems plausible.
5.2.1 Hybrid-QGAN
Until 2019, quantum-GAN research leveraged quantum-classical hybrid algorithms for GAN frameworks almost exclusively [22, 27, 37, 39, 40, 41]. While this work was often referred to as a quantum-GAN, these were HGANs. A typical HGAN design leveraged a VQC for the G, while the D utilized a classical neural network, thereby avoiding the quantum random access memory (QRAM) input bottleneck associated with encoding the real classical data into a quantum state.
The VQC’s connection topology, where qubits were directly connected to the QPU, contributed to HGAN success since variables could be added or tuned, for both the D and G during training [37, 39, 40, 41]. Other HGAN research leveraged binary encoding [41] and amplitude encoding [39] for classical-quantum data preparation, which required considerable resources for the QML computation. A final QC-inspired unitary transformation learning approach [41] attempted classical-quantum data loads using fewer resources.
The VQC flexibility for G-tuning helped to ensure HGAN convergence to a Nash equilibrium. The QC further helped to overcome many of the CGAN training stability challenges [22, 39]. Likewise, HGAN performance times were typically lower than an equivalent CGAN model [22, 27, 40] pushing GAN technology closer to real-world performance expectations. Still, the HGAN’s decreased performance from encoding classical data into a quantum state nullified all QC gains realized [31, 38, 42], thereby requiring exploration into true fully entangled QC GANs.
5.2.2 Fully-entangled QGAN
To extend GAN application fully to QCs, a new QGAN architecture was introduced [31, 38, 42] where both the input and output were comprised entirely of quantum data. This new QGAN design forced all GAN operations onto the QC, bypassing the classical system entirely. As a result, the QML and QRAM performance encumbrances encountered by previous HGANs were eradicated. Unfortunately, bypassing classical systems and forgoing classical-quantum data encoding severely limited QGAN use case applicability since current quantum datasets are infinitesimally small.
This was first illustrated by [42] in their trivial but important QGAN experiment where, using a superconducting transmon qubit, both real and fake data was stored in bosonic modes using an alternating algorithm. Real quantum data was created in an arbitrary state using the bosonic microwave mode, which maintained control of the transmon qubit and the bosonic mode. The QGEN next created the transmon’s fake data from the real quantum state. The QDIS then measured the axis angles of the transmon qubit to determine the ground state. From this point, the traditional GAN adversarial game was played, with gradient measurements calculated to maximize the discriminator for the QDIS’ turn or maximizing the generator for the QGEN’s turn. The adversarial learning process was repeated until either a preset limit was reached or the optimized QDIS discriminator was smaller than a preset threshold. Fidelities as high as 99.1% were achieved for both the real and fake data quantum states. This experiment revealed that a QGAN could achieve convergence without knowing whether the QGEN’s data was real or fake or the QDIS’ selected axis measurement.
A second equally important piece of QGAN research to emerge was [38], who addressed classical-quantum data load limitation issues while building off [42] in their efforts to mitigate HGAN mode collapse issues and convergence errors. Using Google Sycamore, [38] created a novel entangled QGAN that entangled both real and fake QGEN data at the QDIS. This was a deviation from previous QGAN work, which provided the QDIS with either real or fake data, but not both. Moreover, in previous HGAN work, the QGEN supplied a classical discriminator (D) with either the real or the fake sample. Thus, it was impossible to entangle HGAN-QDIS data, an incongruity that routinely caused non-convergence and oscillations due to mode collapse.
This two-qubit entangled QGAN from [38] overcame the mode collapse issues that plagued previous HGAN research and their QGAN converged to a Nash equilibrium. They further showed their QDIS effective at recognizing and suppressing certain quantum errors on Google NISQ devices. While more work was required to determine the feasibility of unsupervised QED/C, [38] showed promise since gradient calculations on QML models are time-consuming, leaving NISQ devices prone to QEs. Lastly, and perhaps most relevant to quantum application development, they demonstrated that their entangled QGAN could create an approximate QRAM for loading classical data in superposition, thereby expediting the classical-quantum data encoding process.
The work of [38] was furthered by [31] who performed three tests of varying hardware and software complexity, against the new entangled QGAN developed by [38]. Google’s Cirq open-source quantum python framework was used by [31] to prepare the VQCs for their experiments, while TensorFlow Quantum (TFQ) was used to add the physical circuits as model layers. Two, four, and six-qubit models were trained using TensorFlow with an Adam optimizer set to a 0.01 learning rate. Noise was then artificially injected into gate rotation angles by adding a gate with a random error after each CNOT, iSWAP, or CZ gate.
The first experiment by [31] consisted of a single layer of randomly selected predefined rotation angles where a simulation performed both a perfect swap test and an adversarial swap test. Their experiments leveraged only the QGEN and thus did not require entangled QGAN training. Results showed the perfect swap test generally performed better. Moreover, the entangled QGAN’s mode was elevated, indicating the data was highly dispersed, a problem notorious to traditional CGANs, since GANs, in general, are difficult to train. Notably, dissimilar rotation angles revealed that while the two amplitudes were identical, the two quantum states were, in fact, different.
The second experiment by [31] challenged the entangled QGAN further, increasing the qubit count and QGEN layers and using real Hamiltonian data produced by a variational quantum eigensolver (VQE). The goal was for the entangled QGAN to learn an unknown Hamiltonian’s approximate eigenstate from the real Hamiltonian. Results in the second experiment showed the QGEN’s adversarial training outperformed the perfect swap test. However, the QDIS’ approximations were off due to the entangled QGAN’s lack of eigenstate phase estimation.
In the third experiment, [31] explored the efficiency of an entangled QGAN learning random states. As the QGEN variables are dynamic, ranging from two, four, or six qubits, and since the VQC only employs k-Nearest Neighbor interactions, [31] expected fidelity to increase parallel to qubit count; however, not all quantum gates performed the same. Instead, certain gates outperformed other gates, with some gates using two or fewer model layers. Thus, certain quantum gates presented a two-qubit gate advantage over simply increasing layers to the TFQ model.
While this new fully entangled QGAN developed by [38] showed promise at overcoming classically hard problems, [31] also noted that challenges rivaling traditional CGAN issues. Likewise, new QC-specific challenges were also presented with this new QGAN model as well. The next and final section of will perform a very simple experiment to test the accuracy and difficulty of the QML research explored throughout this chapter.
This last section will perform a uQGAN experiment using IBM’s Quantum Cloud, IBM Qiskit, IBM’s extensive QC device framework, and Google’s Tensorflow, which is fully supported on the IBM Quantum Cloud. All resources are accessible with minimal configuration, affording maximum development and experimentation time. This uQGAN leveraged IBM’s original QGAN, updating IBM’s now deprecated circuit quantum neural network (CircuitQNN) with IBM’s newest sampler quantum neural network (SamplerQNN) QALGO, introduced May 2023 [43].
The original IBM QGAN lacked a noise model implementation, which was added to this uQGAN to accommodate QED/C experimentation. As such, a backend IBM SamplerAER noise model implementation was added to make it more difficult for the quantum discriminator to detect and correct superconducting-NISQ quantum errors. Lastly, the new uQGAN optimized the Tensorflow hyperparameters for the new SamplerQNN QALGO and noise model. The complete experiment can be retrieved in the Appendix.
This uQGAN will generate the real data from a quantum state. A quantum dataset rather than a classical dataset will be used. This synthetically created quantum mechanical dataset will be generated from a single VQC, with a qubit rotated from an initial ground state of |0⟩ to an arbitrary fixed state. The uQGAN will then be tasked with learning a noise-injected generator circuit from an IBM SamplerAER Noise Model. It will be determined from training statistics, linear regression models, and other visualizations if the QGEN was able to reproduce the same (real data) quantum state for the fake data’s generated quantum state. The goal of this experiment is to determine if a uQGAN is theoretically capable of decreasing QCERs in QIP by mechanizing QED/C.
Table 1 shows the hyperparameters used to train the uQGAN model in this experiment. It is important to remember that GANs comprise two independent models, as such the QGEN and QDIS need to be tuned separately and different optimizers may perform better for one over the other. Here, the best optimizer for the QGEN and QDIS was a stochastic gradient descent (SGD) optimizer, with a Learning Rate of 0.02. Next, 100 epochs, (the number of times all data in the quantum dataset is cycled through the QALGO) was used for the training run. The QGEN to QDIS step rates were set to one and five respectively. Thus, the QDIS updated its model five times for every one QGEN model update. The IBM SamplerAER Noise Model was used for this specific experiment, otherwise, all other hyperparameters were left at their default values.
QGEN
QDIS
Optimizer
SGD
SGD
Learning Rate
0.02
0.02
Epoch
100
100
Steps
1
5
Noise Model
SamplerAER
SamplerAER
Table 1.
uQGAN model hyperparameters.
Table 2 reveals the QGEN and QDIS cost estimates and KL Div. values for the uQGAN noise model independent variables. The lowest QGEN cost estimate for this experiment was −0.375 at Epoch 0, followed closely by −0.382 at Epoch 1. The highest QGEN estimate observed was −0.796 at Epoch 90. In turn, the lowest QDIS cost estimate observed of −0.29 was also at Epoch 0. In turn, the QDIS’ highest cost estimate was −0.206 was encountered at Epoch 90. The largest KL Div. value between the QDIS and QGEN was 1.41 between Epochs 40 and 60, while the smallest KL Div. value was 1.33 at Epoch 0.
Epoch
Generator cost
Discriminator cost
KL Div.
0
−0.375
−0.454
1.33
10
−0.382
−0.585
1.36
20
−0.427
−0.57
1.39
30
−0.487
−0.514
1.4
40
−0.549
−0.453
1.41
50
−0.609
−0.393
1.41
60
−0.666
−0.336
1.41
70
−0.716
−0.286
1.39
80
−0.759
−0.243
1.38
90
−0.796
−0.206
1.35
Table 2.
Kullback-Leibler divergence (KL div.) measurement between the generator and discriminator.
The cost function is a measurement of how accurately the QGEN and QDIS models were in their ability to estimate their relationship between predicted values and their actual values. In turn, the KL Div value measures the distance between the uQGAN model’s real and fake distribution. The lower the KL Div. value the higher the distribution’s similarity. As such, a KL Div. value of zero indicates a distribution is equivalent. Therefore, the KL Div. values below indicate the QGEN and QDIS in this experiment are not equivalent.
Figure 1 shows the uQGAN training results for the QGEN and QDIS linear regression models. The visualization clearly shows the uQGAN model converge when the QGEN (blue) line intersects the QDIS (red) line around the −0.5 Loss mark. However, these results must be correlated against the KL. Div. values, in the subsequent line graph, which does not follow a zero trajectory.
Figure 1.
Converged uQGAN noise model with KL div. and epoch.
Next, Figure 2 shows that the trained quantum data distributions created by the uQGAN’s QGEN are not equal to the real data distribution of a theoretical QC. This is visually apparent by reviewing the register mappings between the trained generator’s distribution and the real distribution, as they are not identical. The trained generator distribution is the IBM QC device on the left, and while only qubits 00 and 11 were utilized by the program, two additional qubits were allocated to account for QEs, these are qubits 01 and 10. These will be excluded from the discussion below.
Figure 2.
Trained generator data distribution verses a real distribution from a perfect theoretical quantum computer.
Next, regarding the quasi-probability of QIP success (00 or 11) verses QE (01 or 10) on the IBM QC, register 11 has the lowest quasi-probability rating at 0.042, while register (or qubit) 00 has the highest quasi-probability rating of 0.419. This is contrary to the theoretical QC which maintains a perfect quasi-probability score of 0.5 for both registers 00 and 11. These results indicate that the uQGAN failed to learn a VQC possessing an unknown state and reproduce that learned state on entangled qubits.
One additional result worth pointing out in Figure 2, with nearly all quantum data placed in qubit 00, a uQGAN mode collapse also certainly occurred. The trained generator’s distribution is not balanced between the two quantum registers 00 and 11. As such, with nearly all the quantum data residing in qubit 00, the QDIS likely became too good at guessing the data.
This chapter commenced with a liberal overview of QIP, featuring QEs common to NISQ devices and the current lack of effective QED/C techniques for decoherence. Quantum error correction research was next discussed, with specific attention given to QML and its potential for QCER reduction. Quantum algorithms were next introduced, setting the groundwork for a thorough discussion of quantum GANs, both hybrid and fully entangled, with special attention given to uQGANs and their perceived applicability to future QECC strategies. The chapter concluded with a uQGAN Noise Model experiment compliments of IBM Quantum Cloud.
There were several key take-aways from this experiment, first, while the uQGAN did converge, closer inspection of the training statistics revealed that the KL Div. values were too high, indicating that the range between the QGEN and QDIS was too great, and the predictions were likely invalid. Thus, the model results were either inaccurate or simply occurred by chance.
Next, the trained generator’s (fake) data distribution did not mirror the real distribution of a perfect QC. Instead, it indicated that the uQGAN was unlikely to mitigate detect and correct QEs by learning a VQC with an unknown state and reproducing its learned state on entangled qubits. Additionally, the observation of quantum data in register 11 was indicative of a mode collapse, indicating the discriminator likely became too good at guessing one type of data.
A great deal of content was covered in this chapter. It was intended to convey the immense amount of work that has gone into furthering the field of Quantum Machine Learning over the past several years. Quantum scale does not exist in a vacuum, it will not be achieved without quantum speed, which will not happen without mechanized quantum error correction.
The material used to create this chapter was derived from the author’s Quantum Machine Learning Doctoral Dissertation. This work, however, was specifically crafted for the quantum-curious crowd, rather than the astute quantum research community.
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Written By
Theresa Melvin
Submitted: 30 August 2023Reviewed: 04 September 2023Published: 05 December 2023
Open access peer-reviewed chapter
