Summary: Chicago Quantum Exchange (CQE) Pulse-level Quantum Control Workshop (2024)

III-A Variational Algorithms

Variational quantum algorithms (VQA) provide an exciting opportunity for near-term machines to demonstrate quantum advantage. VQAs have a one-to-one mapping between logical qubits in algorithms to physical qubits in QCs. Physical qubits will be discussed more in SectionIV-A. Proposed applications of VQAs include ground state energy estimation of molecules[4] and MAXCUT approximation[2]. VQAs are hybrid algorithms since they comprise of classical and quantum subroutines.

VQAs are well suited for near-term hardware since they adapt to the intrinsic noise properties of the QC they run on. The VQA circuit is parameterized by a vector of angles that correspond to gate rotations on qubits. The vector of angles is optimized by a classical optimizer during many iterations to either maximize or minimize an objective function that represents the problem that the VQA implementation hopes to solve.

Two notable VQAs are the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA). The former is often used for quantum chemistry while the latter is applied to combinatorial optimization.

III-B Fault Tolerant Algorithms

Fault tolerant quantum algorithms offer exciting computational gains over their classical analogs, but they require extremely low error thresholds. Because of this, error correcting codes (ECCs) are required that enforce a one-to-many mapping between logical qubits in an algorithm to physical qubits on a QC. This encoding allows quantum information to be shielded from errors that arise due to uncontrolled coupling with the environment or due to imperfect operations at the physical level. There are many kinds of quantum ECCs[5].

At minimum, it is estimated that millions of high-quality physical qubits will be needed to implement fault-tolerant quantum computation[6]. Once fault tolerance is reached, disruptive implementations of Shor’s algorithm for quantum factoring[7], Grover’s quantum database search[8], and quantum phase estimation[9] can be implemented.

IV-A Physical Qubits

Even if the theory is well defined, information must have a physical realization for computation to occur. Currently, a variety of quantum technologies can encode logical qubits within different media. We call a physical implementation of a radix-2 unit of information a physical qubit. Since today’s quantum devices lack error correction, each logical qubit within an algorithm is implemented with one physical qubit in a machine. These devices, sometimes called Noisy Intermediate Scale Quantum (NISQ) devices, are error prone and are up to hundreds of qubits in size[10]. Some physical qubit examples include energy levels within superconducting circuits[11, 12, 13], ions trapped by surrounding electrodes[14, 15, 16], neutral atoms held with optical tweezers[17, 18], and photons travelling through free space or waveguides[19, 20]. Each of these platforms have their unique strengths, but none has become the obvious choice for the standard quantum computing platform.

IV-B Physical Architectural Constraints

Although many promising quantum technologies exist, they are imperfect and demonstrate attributes that prevent scaling in the near-term. For example, quantum state preparation, gate evolution, and measurement all are characterized by nontrivial infidelity rates and must be implemented with a limited set of instructions that depend on the control signals that are available for a specific qubit technology. Additionally, restricted connectivity among device qubits is a key limiting factor. Qubit-qubit interaction is necessary for quantum entanglement, but unfortunately, many of today’s QCs are limited in the number of qubits that are able to interact directly. Even with technologies that allow for all-to-all connectivity, such as with trapped ions, QCs are limited by the total number of qubits that are actively used during computation[21]. Careful design of qubit layout on devices as well as intelligent algorithm mappers and compilers can assist with qubit communication overheads.

Qubit communication is required for quantum algorithms, but unintended communication, or crosstalk, can cause errors during computation. Crosstalk errors experienced during quantum algorithm execution often arise from simultaneously stimulating two neighboring qubits. If the qubits are strongly coupled and “feel” each other, the fidelity of the two simultaneous computations can decrease. Fortunately, qubit crosstalk errors are systematic. Once identified, sources of crosstalk can be mitigated with software[22, 23] or with improved device design[24].

Today’s qubits are extremely sensitive, and as a result, crosstalk with neighboring qubits is not the only accidental signal interference that must be considered for quantum systems. Qubits also tend to couple with their surrounding environment, causing quantum information to decohere. Amplitude damping and dephasing cause qubit decoherence errors. Amplitude damping describes the loss of energy from $\ket{1}$ to $\ket{0}$, and $T_{1}$ is used to denote the exponential decay time from an excited qubit state to a ground state. Dephasing error refers to the gradual loss of phase between $\ket{0}$ to $\ket{1}$ and is described by the time $T_{2}$. Quantum decoherence sets rigid windows for compute time on current QCs. If the qubit runtime, or period spanning the first gate up until measurement, surpasses the QC $T_{1}$ or $T_{2}$ time, the final result of computation may resemble a random distribution rather than the correct output.

IV-C Superconducting Circuits - A Case Study on Progress

Superconducting circuits have emerged as a leading physical qubit implementation[25]. IBM has dedicated much research to the development of the Transmon-based superconducting QC, and since 2016, the company has allowed their prototype devices to be used by the public via the IBM Quantum Experience[26]. Since 2016, IBM QCs have increased from 5 to 127 qubits[27]. The significance of increasing the number of qubits on-chip to a total of $n$ is that more complex algorithms can be run that explore an exponentially larger state space, dimension $2^{n}$. Although the debut of larger QCs with over a hundred qubits is an impressive engineering development, limited fidelity, qubit communication, and coherence times prohibit these devices from reliably executing circuits that require all of a device’s physical qubits to work in synergy.

Progress in quantum hardware is not measured simply by the consistency and count of device qubits. The gate error must also be considered as it influences the depth of circuits that a QC can run. IBM QCs have seen the average two-qubit infidelity decrease in magnitude from order $10^{-2}$ to $10^{-3}$ over the last five years[27]. Quality of multi-qubit gates is an important metric to consider when rating a QC because it enables the entanglement required for many quantum algorithms to demonstrate advantage. The importance of device size, qubit quality, and operation success is clear, thus, improving quantum systems as a whole requires a multi-targeted approach. To assist with scaling into more robust devices, IBM developed a metric referred to as Quantum Volume (QV) that aims to benchmark quantum hardware in the near term. QV is a scalar value, where higher is better, that assesses the largest random circuit of equal width and depth equal to $m$ that a QC can successfully run[28]. As QV$=2^{m}$, the metric provides perspective on system performance as a general purpose QC. Capability in terms of coherence, calibration, state initialization, gate fidelity, crosstalk, and measurement quality are all captured by QV. QV is also influenced by design aspects such as chip topology, compilation tools, and basis gate set.

Recently, IBM extended the QC frontier to achieve a QV of 64, and then later 128, through the combined application of improved compiler optimizations, shorter two-qubit gates, excited state promoted readout, higher-accuracy state discrimination and open-loop correction via dynamical decoupling[13, 29]. These breakthroughs were accompished on 27 qubit devices, demonstrating that the compute power of a quantum system is heavily influenced on compilation and low-level control; it does not depend on qubit count alone. Quantum compilation and low-level control will be discussed further in SectionV. By focusing research to innovate in these domains, along with making developments in other areas of QC architecture, IBM anticipates to scale QCs to one million qubits and beyond[30].

V-A Qiskit and the Pulse Programming Model

Qiskit is an open-source quantum computing framework from IBM that provides tools for creating, manipulating, and running quantum programs on quantum systems independent of their underlying technology and architecture.

The commonly used quantum programming paradigm is the circuit model.Such a model abstracts the physical execution of a quantum algorithm on a quantum system into a sequence of unitary gate operations on a set of qubits followed by qubit measurements.The gates manipulate the qubit states while measurements project these qubits onto a particular measurement basis that are extracted as classical bit-strings.Qiskit supports this programming model via a quantum assembly language called OpenQASM[40]. OpenQASM is simply an abstraction of the real quantum hardware execution in a form which is amenable to users familiar with classical programming models.The hardware is not capable of naively implementing or executing the quantum instructions from this model and must instead compose these operations via the control hardware.

At the device level, quantum system execution is implemented by steering the qubit(s) through a desired unitary evolution, which is achieved by careful engineering of applied classical control fields.Thus, execution of a program represented via the quantum circuit model on a quantum system requires translating or compiling gate-level circuit instructions to a set of microwave control instructions, or pulses, which enact the desired state-transformations or measurements.This translation is often suboptimal due to the heavy abstractions imposed across the software stack. In the circuit programming model, an atomic circuit instruction is agnostic to its pulse-level implementation on hardware, and unfortunately, the vast majority of program optimization is often done at the gate-level in the standard circuit model.Extracting the highest performance out of quantum hardware would require the ability to craft a pulse-level instruction schedule for the optimization of circuit partitions.

Qiskit Pulse[38, 39] was developed to describe quantum programs as a sequence of pulses, scheduled in time.Qiskit Pulse adds to the Qiskit compilation pipeline the capability to schedule a quantum circuit into a pulse program intermediate representation, perform analysis and optimizations, and then compile to Qiskit Pulse object code to execute on a quantum system.

To program such systems at the pulse-level in a hardware-independent manner requires the user-level instruction set to be target-compiled to the underlying system hardware components, each of which may have a unique instruction set and programming model.Qiskit Pulse provides a common and reusable suite of technology-independent quantum control techniques that operate at the level of an analog stimulus, which may be remotely re-targeted to diverse quantum systems.

Fig.1 shows an example of how a pulse schedule can be explicitly coded in Qiskit. While this is a trivial example, readers can refer to [39] for an example of implementing a high-fidelity $CX$ gate based on the calibrated Cross-Resonance (CR) pulse. The CR gate is an important microwave-activated, two-qubit entangling operation that is performed by driving one qubit, the control, by the frequency of another, the target[41].

Fig.2 shows provides a code snippet for scheduling a quantum circuit into a pulse schedule using Qiskit.Qiskit Pulse users may create pulse programs to replace the default pulse programs of the native gate set provided by the backend and pass them as an argument to the scheduler.

V-B Pulse Support via OpenQASM3

OpenQASM has become a de facto standard, allowing a number of independent tools to interoperate using OpenQASM as the common interchange format.While OpenQASM2 uses a circuit model which was described previously, quantum paradigms require going beyond the circuit model, incorporating primitives such as teleportation and the measurement model of quantum computing.OpenQASM3[40] describes a broader set of quantum circuits with concepts beyond simple qubits and gates.Chief among them are arbitrary classical control flow, gate modifiers (e.g. control and inverse), timing, and microcoded pulse implementations.

To use the same tools for circuit development as well as for the lower-level control sequences needed for calibration, characterization, and error mitigation, it is necessary tocontrol timing and to connect quantum instructions with their pulse-level implementations for various qubit modalities.This is critical for working with techniques such as dynamic decoupling[42, 43] as well as for better characterization of decoherence and crosstalk.These are all sensitive to time and can be programmed via the timing features in OpenQASM3.One such potential application compilation and execution flow is shown in Fig.3.

To control timing, OpenQASM3 introduces “delay” statements and “duration” types.The delay statement allows the programmer to specify relative timing of operations.Timing instructions can use the duration type which represents amounts of time measured in seconds.

OpenQASM3 includes features called “box” and “barrier” to constrain the reordering of gates, where the timing of those gates might otherwise be changed by the compiler. Additionally, without these directives, the desired gates could be removed entirely as a valid optimization on the logical level.

OpenQASM3 also allows specifying relative timing of operations rather than absolute timing.This allows more flexible timing of operations, which can be helpful in a setting with a variety of calibrated gates with different durations.To do so, it introduces a new type “stretch”, representing a duration of time which is resolvable to a concrete duration at compile time once the exact durations of calibrated gates are known. This increases circuit portability by decoupling the circuit timing intent from the underlying pulses, which may vary from machine to machine or even from day to day.

OpenQASM3 also has added support for specifying instruction calibrations in the form of“defcal,” short for ‘define calibration,’ declarations which allow the programmer to specify amicrocoded implementation of a gate, measure, or reset instruction.

While only a few features of OpenQASM3 are discussed here, the overall design intends to be a multilevel intermediate representation (IR), where the focus shifts from target-agnostic computation to a concrete implementation as more hardware specificity is introduced.An OpenQASM circuit can also mix different abstraction levels by introducing constraintswhere needed, but allowing the compiler to make decisions where there are no constraints.

VI-A Optimized Compilation of Aggregated Instructions for Realistic Quantum Computers

The work[44] proposes a quantum compilation technique that optimizes for pulse control by breaking across existing abstraction barriers.Doing so reduces the execution latency while also making optimal pulse-level control practical for larger numbers of qubits.Rather than directly translating one- and two-qubit gates to control pulses, the proposed framework first aggregates these small gates into larger operations.Then the framework manipulates these aggregates in two ways. First, it finds commutative operations that allow for much more efficient schedules of control pulses. Second, it uses quantum optimal control on the aggregates to produce a set of control pulses optimized for the the underlying physical architecture.In all, the technique greatly exploits pulse-level control, which improves quantum efficiency over traditional gate-based methods.At the same time, it mitigates the scalability problem of quantum optimal control methods.Since the technique is software-based, these results can see practical implementation much faster than experimental approaches for improving physical device latency.Compared to traditional gate-based methods, the technique achieves execution mean speedup of 5x with a maximum speedup of 10x.

Two novel techniques are implemented:a) detecting diagonal unitaries and scheduling commutative instructions to reduce the critical path of computation;b) blocking quantum circuits in a way that scales the optimal control beyond 10 qubits without compromising parallelism.

For quantum computers, achieving these speedups, and thereby reducing latency, is do-or-die: if circuits take too long, the qubits decohere by the end of the computation.By reducing the latency 2-10x, this work provides an accelerated pathway to running useful quantum algorithms, without needing to wait years for hardware with 2-10x longer qubit lifetimes.An illustrative example is shown in Fig.4.

VI-B Partial Compilation of Variational Algorithms for Noisy Intermediate-Scale Quantum Machines

Each iteration of a variational algorithm depends on the results of the previous iteration–hence, the compilation must be interleaved through the computation.The noise levels in current quantum machines and the complexity of the variational use cases that are useful to solve, result in a very complex parameter tuning space for most algorithms.Thus, even small instances require thousands of iterations.Considering that the circuit compilation is on the execution critical path and cannot be hidden, the compilation latency for each iteration becomes a serious limitation.

To cope with this limitation on compilation latency, past work on VQAs has performed compilation under the standard gate-based model.This methodology has the advantage of extremely fast compilation – a lookup table maps each gate to a sequence of machine-level control pulses so that the compilation simply amounts to concatenating the pulses corresponding to eachgate.

The gate-based compilation model is known to fall short of the GRadient Ascent Pulse Engineering (GRAPE)[46, 47] compilation technique, which compiles directly to the level of the machine-level control pulses that a QC actually executes.As has been the theme of this paper, the pulse-level control provides for considerably more efficient execution on the quantum machine.GRAPE has been used to achieve 2-5x pulse speedups over gate-based compilation for a range of quantum algorithms, resulting in lower decoherence and thus increased fidelity.

However, GRAPE-based compilation has a substantial cost: compilation time.This would potentially amount to several weeks or months of total compilation latency during thousands of iterations since millions of iterations will be needed for larger problems of significance. By contrast, typical pulse times for quantum circuits are on the order of microseconds, so the compilation latency imposed by GRAPE is untenable.

This proposal[45] introduces the idea of partial compilation, a strategy that approaches the pulse duration speedup of GRAPE, but with a manageable overhead in compilation latency.This powerful new compiler capability enables a realistic architectural choice of pulse-level control for more complex near-term applications.Two variations are proposed: a) strict partial compilation, a strategy that pre-computes optimal pulses for parametrization-independent blocks of gates; and b) flexible partial compilation, a strategy that performs as well as full GRAPE, but with a dramatic speedup in compilation latency via precomputed hyperparameter optimization.An illustration of the flexible partial compilation is shown in Fig.5.

VI-C Optimized Quantum Compilation for Near-Term Algorithms with Qiskit Pulse

While pulse optimization has shown promise in previous quantum optimal control (QOC), noisy experimental systems are not entirely ready for compilation via QOC approaches.This is because QOC requires an extremely accurate model of the machine, i.e., its Hamiltonian.Hamiltonians are difficult to measure experimentally and moreover, they drift significantly between daily recalibrations.

Experimental QOC papers incur significant pre-execution calibration overhead to address this issue.By contrast, this work[48], [49] proposes a technique that is bootstrapped purely from daily calibrations that are already performed for thestandard set of basis gates.The resulting pulses are used to create an augmented basis gate set.These pulses are extremely simple, which reduces the control error and also preserves intuition about underlying operations, unlike traditional QOC.This technique leads to optimized programs, with mean 1.6x error reduction and 2x speedup for near-term algorithms.The proposed approach can target any underlying quantum hardware.An overview is shown in Fig.6

Four key optimizations are proposed, all of which are enabled by pulse-level control:(a) access to pulse-level control allows implementing any single-qubit operation directly with high fidelity, circumventing inefficiencies from standard compilation;(b) although gates have the illusion of atomicity, the true atomic units are pulses. The proposed compiler creates new cancellation optimizations that are otherwise invisible;(c) Two-qubit operations are compiled directly down to the two-qubit interactions that the hardware actually implements;(d) Pulse control enables $d$-level qudit operations, beyond the 2-level qubit subspace.

VII-A Classical Simulation: Capturing the Time-varying Nature of Open Quantum Systems

Classical simulation of quantum devices is critical to better understand device behavior. This is especially true in the near-term for noisy prototype architectures. Simulation tools accomplish a variety of different tasks including modeling noise sources, validating calculations and designs, and evaluating quantum algorithms. Since quantum simulation tools are classical, they often require supercomputers to simulate quantum devices that are large in terms of either qubit count or hardware components they contain. Quantum simulators imitate the operation of quantum devices and can be used to build the Hamiltonians required to derive control pulses for the qubit or qudit space. As a result, quantum simulators are an important tool in control engineering because complex quantum operations require the generation of fine-tuned drive signals.

Quantum systems are dynamic and evolve with time. As a result, simulation tools for quantum mechanical systems need to take variation into consideration, whether that variation stems from intentional gate application or from unintentional drift or environmental coupling. QuaC, or “Quantum in C,” was developed to capture the time-varying nature of open quantum systems, including realistic amplitude damping and dephasing along with other correlated noises and thermal effects[50]. The simulator is not limited to a specific type of qubit or qudit technology, and although QuaC is classical simulator, the tool can model quantum systems with enough granularity to develop pulses required for low-level control. QuaC has been used for many applications, such as in the discovery of improved error models to better understand noisy quantum systems[51] and for the study of the coupling required between quantum dots and photonic cavities for entanglement transfer[52]. Additionally, QuaC was applied in the comparison of different quantum memory architectures to discover optimum features, such as encoding dimension[53].

VII-B Quantum Simulation: Hardware Efficient Simulation with Qiskit Pulse

The proposal in[54] simulates a quantum topological condensed matter system on an IBM quantum processor.The simulation is done within the qubits’ coherence times using pulse-level instructions provided by Qiskit Pulse.Ideally, capturing the system characteristics would require simulation by continuous time-evolution of qubits under the appropriate spin Hamiltonian obtained from a transformation of the fermion Hamiltonian.However, practically, this is run on quantum devices by having this “analog” simulation decomposed and mapped onto the calibrated native basis gates of a quantum computer, making it “digital”.This digital implementation on noisy quantum hardware limits precision and flexibility and is thus unable to avoid the accumulation of unnecessary errors.Pulse-level control improves this by allowing for a “semi-analog” approach.The proposal shows a pulse-scaling technique that, without additional calibration, gets closer to the ideal analog simulation.

Topologically-protected quantum computation works by moving non-Abelian anyons, such as Majorana zero modes (MZMs), around each other in two dimensions to form three-dimensional braids in space-time[55].Thus far, there has been no definitive experimental evidence of braiding due to dynamical state evolution[56]

This work simulates a key part of a topological quantum computer: the dynamics of braiding of a pair of MZMs on a trijunction.Braiding is implemented by parametrically adjusting the Hamiltonian parameters; the time evolution is implemented using the Suzuki-Trotter decomposition, with each time step implemented by one- and two-qubit gates.Fidelity is significantly improved by using pulse-level control to scale cross resonance (CR) gates derived from those pre-calibrated on the backend, thereby enabling coupling of qubit pairs with shorter CR gate times.Specifically, using native $CX$ gates, only 1/6th of a full braid can be performed.Whereas, using the pulse-enabled scaled gates leads to performing a complete braid.

VIII-A Error-robust Single-qubit Gate Set Design

The work[57] proposes analog-layer programming on superconducting quantum hardware to implement and test a new error-robust single-qubit gate set.To build this, analog pulse waveforms are numerically optimized via the use of Boulder Opal, a custom Tensorflow-based package[58].The optimized pulses enact gates which are more resilient against dephasing, control-amplitude fluctuations, and crosstalk. Before the implementation in real hardware, these pulses go through a calibration protocol that can be fully automated[59] to account for small distortions that may happen in the control channels.The experiments are performed on IBM quantum machines and programmed via the Qiskit Pulse API, which translates the pulses designed using Boulder Opal into hardware instructions.

The experiments show that when pulses are optimized to be robust against amplitude or dephasing errors, they can outperform the default calibrated Derivative Removal by Adiabatic Gate (DRAG) operations under native noise conditions.These optimized pulses are built including both a 30-MHz-bandwidth sinc-smoothing-function and temporal discretization to match hardware programming.

Using optimized pulses, single-qubit coherent error rates originating from sources such as gate miscalibration or drift are reduced by up to an order of magnitude, with average device-wide performance improvements of 5x.The same optimized pulses reduce the gate error variability across qubits and over time by an order of magnitude.

VIII-B Reinforcement Learning for Error-Robust Gate Set Design

As discussed above, it has been demonstrated that the use of robust and optimal control techniques for gateset design can lead to dramatic improvements in hardware performance and computational capabilities.

The design process is straightforward when Hamiltonian representations of the underlying system are precisely known, but is considerably difficult in state-of-the-art large-scale experimental systems.A combination of effects introduces challenges not faced in simpler systems including unknown and transient Hamiltonian terms, control signal distortion, crosstalk, and temporally varying environmental noise.In all cases, complete characterization of Hamiltonian terms, their dependencies, and dynamics becomes unwieldy as the system size grows.

The work[60] proposes a black-box approach to designing an error-robust universal quantum gate set using a deep reinforcement learning (DRL) model, as shown in the inset of Fig.7(a).The DRL agent is tasked to learn how to execute high fidelity constituent operations which can be used to construct a universal gate set.It iteratively constructs a model of the relevant effects of a set of available controls on quantum computer hardware, incorporating both targeted responses and undesired effects.It constructs an $RX$($\pi/2$) single-qubit driven rotation and a $ZX$($-\pi/2$) multi-qubit entangling operation by exploring a space of piecewise constant (PWC) operations executed on a superconducting quantum computer programmed using Qiskit Pulse.

The constructed single-qubit gates outperform the default DRAG gates in randomized benchmarking with up to a 3x reduction in gate duration.Furthermore, the use of DRL defined entangling gates within quantum circuits for the SWAP operation shows 1.45x lower error than calibrated hardware defaults.These gates are shown to exhibit robustness against common system drifts, providing weeks of performance without needing intermediate recalibration.

IX-A Robust Quantum Control over Cavity-Transmon Systems

Oscillator cavities with photonic or phononic modes have long coherence times extending to tens of milliseconds, but unfortunately, these devices are difficult to control, prohibiting the generation of arbitrary quantum states needed for quantum computation[61]. However, if the storage cavity is coupled directly to a Transmon qubit, quantum states can be prepared in the Transmon and swapped into the storage cavity for universal control. After the quantum SWAP occurs, the oscillator holds a bosonic qubit. This idea led to the development of the superconducting cavity qubit module.

Superconducting cavity qubit modules are built using superconducting circuits comprised of a Transmon qubit, a storage cavity, and a readout cavity connected to a coupler port [62, 63, 64]. In this system, the Transmon qubit is used for quantum information processing, while the storage cavity, or the oscillator, can interact with the Transmon to encode state within an oscillator mode[65]. These systems are characterized by a long-lived superconducting storage cavity coherence, a large Hilbert space for representing states, and fast, high-fidelity measurement and readout of the qubit state. All of these features make cavity qubit modules an attractive choice for storing encoded qubits. They also have potential within future quantum error correction protocols.

The work in[65, 61] presents the SNAP gate that improves previous efforts of Transmon-cavity interaction for quantum computation. This gate efficiently enables the construction of arbitrary unitary operations, offering a scalable path towards performing quantum computation on qubits encoded in oscillators. Combined cavity and Transmon drive pulses provide control of the system to implement the SNAP gate. These pulses can be generated by gradient-based optimum control, or GRAPE[66].

The superconducting cavity module consisting of a storage cavity and a Transmon qubit is an example of an ancilla-assisted system. The main goal of the system to take advantage of the long coherence times of oscillator modes. The challenge of limited control associated with the storage cavity is alleviated through Transmon qubit coupling. Employing a Transmon in the cavity qubit module, however, does not come without cost. The ancilla qubit injects noise to the system due to shorter coherence windows associated with the Transmon (tens of microseconds). Thus, there is room to further improve the quantum control of the superconducting cavity module. Quantum error correction techniques are required to protect cavity systems from ancilla-introduced errors. Fortunately, a solution based on path independence has been proposed to develop fault tolerant quantum gates that are robust to ancilla errors[67, 68].

IX-B Specialized Architectures for Error Correction

Quantum control allows the desired quantum device behavior from carefully designed classical signals. Control techniques that continue to reduce error in near term QCs are still under development, but it must be kept in mind that the goal of quantum science is to eventually build architectures that can implement fault tolerance. As a result, new device architectures and supporting control systems should be designed concurrently.

Looking forward, there are many design bottlenecks that must be addressed to scale the current qubit technology. For example, superconducting qubits face issues associated with crosstalk between qubits, limited area for control wires, and inconsistencies during fabrication. These challenges must be sidestepped, and one approach is to reduce the amount of hardware scaling required to minimize the burden on engineering and material science efforts.

New architectures based on superconducting cavity technology have been proposed that aim to implement fault tolerance while reducing the requirements of the physical hardware[69]. With reduced hardware, some of the challenges associated with scaling quantum control mechanisms will be resolved.

The approach in[69] aims to move towards scalable fault tolerant architectures by combining compute qubits with memory qubits for a 2.5D device design. The combined compute and memory unit will be constructed with a Transmon coupled with a superconducting cavity. The cavity is characterized by coherence times that are much longer than those of the Transmon qubit. This advantage allows the cavity memory unit coupled to the Transmon compute unit to be used for random access to error-corrected, logical qubits stored across different memories, creating a simple virtual and physical address scheme. Error correction is performed continuously by loading each qubit from memory. The error correction used in[69] includes two efficient adaptations of the surface code, Natural and Compact.

The 2.5D architecture has many architectural advantages that provide avenues for simplified control engineering. First, the combined Transmon and cavity module requires only one set of control wires. Compared to traditional 2D architectures that require dedicated control wires and signal generators, there is potential to reduce the amount of control hardware by a factor of $n$ where $n$ is the number of modes, and thus the number of qubits, that can be stored in the storage cavity. Additionally, the 2.5D architecture allows for transversal $CX$ operations that can extend into the z-plane of the cavity to act on qubits stored within modes. This $CX$ operation allows for lattice surgery operations with improved connectivity between logical qubits and that can be executed 6x faster than standard lattice surgery $CX$ operations. Faster gates are a huge win for device control when qubits are constrained by a coherence time.

X-A Toffoli Gate Depth Reduction in Fixed Frequency Transmon Qubits

Quantum information encoding is typically binary, but unlike classical computers, most QCs have multiple accessible energy levels beyond the lowest two used to realize a qubit. Accidental use of these upper energy levels can insert errors into computation. However, these energy levels that transform a qubit into a qudit can be intentionally used with careful quantum control to achieve performance gains during computation. Control pulses that access the qudit space must be carefully designed, taking hardware constraints such as shorter decay time associated with higher energy levels into consideration[70].

Implementing high-dimensional encoding for quantum information provides the benefit of data compression since qubit states can be represented with qudit formalism. Condensed information storage with qudits has been proposed for use in efficient applications of quantum error correction[71], communication protocols[72], and cryptography[73]. Another exciting application for high-dimensional encoding is in gate depth reduction for multi-qubit gates[74, 75]. Multi-qubit gates are required in many quantum algorithms, but they must be decomposed into smaller operations that agree with both the basis gate library and connectivity graph of a QC. These decompositions can be costly in terms of total single- and two-qubit operations, so opportunities for gate depth reduction can directly improve the overall circuit fidelity on today’s noisy QCs.

Qiskit Pulse allows users to access of higher energy levels on their Transmon-based QCs[76]. Using this low-level control of the hardware, gates for qutrits, or radix-3 quantum information that uses the basis states of $\ket{0}$, $\ket{1}$, and $\ket{2}$, can be defined. Recent work using Qiskit Pulse experimentally demonstrated that extending into the qutrit space can provide advantages for qubit-based computing[77, 78]. In these schemes, qudits were used for intermediate computation in gate realization. The overall algorithm, however, still processes in radix-2, maintaining its “qubits in, qubits out” structure.

In[78], the authors propose a Toffoli gate decomposition that is improved by intermediate qutrits. The Toffoli gate, or controlled-$CX$ ($CCX$), is an important quantum gate that has been targeted by optimal control techniques since it is widely used in reversible computation, error correction, and chemistry simulation among other quantum applications[79]. The authors introduce a decomposition that uses single- and two-qutrit gates to achieve an order-preserving, Toffoli decomposition that only requires four, two-Transmon interactions. The value of being order preserving is that the operation is friendly to near-term devices that demonstrate nearest-neighbor connections.

Ref.[78] describes the control pulses required to realize the qutrit operations on IBM hardware. The successful demonstration of this Toffoli decomposition provides a significant gate reduction from the optimum qubit-based alternative that requires eight, two-Transmon interactions. Average gate fidelity of qutrit Toffoli execution was measured to be about 78%. The qutrit Toffoli gate was benchmarked against the optimum qubit-based decomposition showing a mean fidelity improvement of around 3.82% and an execution time reduction of 1$\mu s$. Additional error correction techniques via quantum control measures, such as dynamical decoupling, were also employed for further performance gains with the qutrit Toffoli.

X-B High-dimensional GHZ Demonstration

Quantum entanglement is a hallmark that distinguishes quantum computation from classical processing techniques. It has a wide range of applications from secure communication protocols to high-precision metrology. There have been many experimental demonstrations of entanglement, but they have primarily focused on radix-2 systems. An advantage of exploring higher-dimensional entanglement includes increased bandwith in quantum communication protocols, such as those for superdense codes and teleportation.

At least two systems are needed for quantum entanglement, and a GHZ state is a special type of entanglement, called multipartite entanglement, that is shared between three or more subsystems, i.e. qubits or qudits. High-dimensional entanglement has been demonstrated on numerous occasions with photonic devices[80], but with careful pulse-level control using Qiskit Pulse, a qutrit GHZ state shared between three qudits was prepared on IBM Transmon QC[81]. This demonstration was the first of its kind on superconducting quantum technology.

In[81], the GHZ circuit for three qutrits was built using IBM calibrated gates, $RY(\theta)$ and $CX$, along with specially programmed and calibrated single-qutrit gates. To detect qutrit states, a custom discriminator was developed that is sensitive to three basis states for during measurement. Experiments to generate entanglement were ran using IBM Quantum Cloud resources, and GHZ states were produced around 30000 times faster than leading photonic experiments[82]. Tomography confirmed the fidelity of entangled state preparation on a five-qubit IBM machine to be approximately 76%. Additionally, the three-qutrit GHZ state was verified further using the entanglement witness protocol.

XI-A Quantum Hardware Education

Reaching fault tolerant QCs is heavily dependent on dedicating adequate resources to develop improved hardware and control. This involves building a quantum community with skill sets that range from quantum aware to quantum specialist. A first step to expanding the quantum hardware community is limiting the barriers to entry. The quantum net could be cast to a wider audience by increasing efforts to introduce concepts related to quantum computing and technology at all education levels. Additionally, expanding quantum optimal control research will include engaging a diverse set of willing students of all ages. STEM has largely overlooked many demographic groups, but as quantum research is still in its infancy compared to many classical fields, there is opportunity to ensure that engineers and scientists from underrepresented groups feel a sense of belonging in the quantum community and feel that they can contribute to its growth. Expanding diversity and inclusion in quantum computing will continue to advance the field in both the academic and industrial setting.

In this Section, we will discuss three key strategies to allow quantum information to reach a wider audience by tailoring instruction to different levels of education. First, if students become more familiar with core quantum concepts at an early age, they are better prepared to pursue careers in quantum hardware and quantum control. Second, as the demand for a quantum workforce increases, higher education will gain a better perspective of what is required for degree plans in quantum science at both the undergraduate and graduate levels. Finally, it is possible to train seasoned scientists and engineers from classical areas of study so that their refined expertise can be applied to the research and development of quantum technology.

XI-B Improving Quantum Technology Awareness

Many exciting developments in the field of quantum information processing are unfortunately buried in research papers, specialized conference sessions, and meetings that are inaccessible to those not immediately working on the issues at hand. So while quantum computing holds great potential, there is the risk that the emerging technology could be undervalued or worse, misunderstood. A goal of expanding the quantum hardware community is to increase the public’s awareness about the reasonable expectations and timelines to have for QCs. By improving communication channels and trust between the QIS community and the rest of the world, more individuals may consider how quantum research may impact their lives or even how they can get involved. Initial steps such as research groups producing news articles describing theoretical and experimental breakthroughs in layperson’s terms could make a huge difference so that quantum progress reaches a wider audience is understood with greater accuracy.