Understanding Superconducting Qubits in One Article
The key to making a useful quantum computer is the development of multi-qubit processors. A prominent area of building multi-qubit processors is superconducting qubits, where information is stored in the quantum degrees of freedom (DOF) of nanoscale anharmonic oscillators (AHOs) built from superconducting circuit elements. Specifically, superconducting qubits have the following advantages:
1) High designability. Superconducting qubit systems can be designed with different types of qubits, such as charge qubits, magnetic flux qubits, and phase qubits; and different parameters, such as the energy level and coupling strength of qubits, can also be adjusted by adjusting capacitance, inductance, and Joseph Sen energy to adjust.
2) Scalability. The fabrication of superconducting qubits is based on existing semiconductor microfabrication processes: using advanced chip fabrication techniques, high-quality devices can be fabricated, which is beneficial for both fabrication and scalability.
3) Easy to couple. The circuit properties of superconducting qubit systems make it relatively easy to couple multiple qubits together: in general, coupling can be via capacitance or inductance.
4) Easy to control. The manipulation, measurement of superconducting qubits is compatible with microwave control and operability. Therefore, commercial microwave equipment and equipment can be used for superconducting quantum computing experiments.
How can quantum systems based on superconducting circuits be designed for applications?
1) Quantization of Linear Oscillating Circuits
A quantum mechanical system depends on the time-dependent Schrödinger equation:
Among them, |ψ(t)⟩ is the state of the quantum system at time t, ℏ is the simplified Planck constant h/2p, and H^ is the “Hamiltonian” describing the total energy of the system, which determines the The Hamiltonian is the first step in deriving its dynamical behavior.
Taking the classic description of a dissipative (like superconducting) linear LC resonant circuit as an example, the energy in the system oscillates between the electrical energy in the capacitor C and the magnetic energy in the inductor L. The linear character of harmonic oscillators (QHOs) has natural limitations when dealing with quantum information: since many gate operations depend on frequency selectivity, the QHO potential energy curves shown in Fig. 1b have equidistant horizontal spacing. Specifically, we want to realize qubits through superconducting circuits. For example, take the ground state as |0⟩ and the first excited state as |1⟩. But since the energy levels of the system are equally spaced, there is no guarantee that the qubits are just going from |0⟩ to |1⟩ but not from |1⟩ to |2⟩, or even higher energy levels. So a superconducting LC circuit is not an easily manipulated qubit.
Figure 1 a. Parallel LC oscillator circuit with inductor L in parallel with capacitor C, superconducting phase denoted ϕ, reference ground state zero; b. Energy potential of QHO, where energy levels are equally spaced ℏωr ; c. Josephson Qubit circuit in which the nonlinear inductance LJ (in orange dashed box) is shunted by capacitor Cs ; d. The Josephson inductance reshapes the secondary energy potential (red dashed line) into a sine wave (blue solid line), which produces Asymmetric energy levels, able to isolate the two lowest energy levels |0⟩ and |1⟩, forming an energy-separated ℏω01 operator space, different from ℏω12 .
In order to alleviate this dynamic error problem, anharmonicity (or nonlinearity) needs to be added to the system. Generally speaking, the larger the anharmonicity, the better. Hence, the introduction of the Josephson junction—a nonlinear, dissipative circuit element that forms the backbone of a superconducting circuit. As shown in Figure 2, the form of potential energy can be adjusted by replacing the linear inductance of the QHO by the Josephson junction as a nonlinear inductance.
Figure 2 a. Morphology of Josephson junction; b. Circuit representation of Josephson junction; c. Volt-ampere characteristic curve of Josephson junction
At this time, the modified Hamiltonian equation is:
Among them, EC=e2/(2C ∑ ) , C ∑ =Cs+Cj is the total capacitance; EJ=Ic( Φ0/2π) represents the energy of the Josephson junction. With the introduction of a Josephson junction in a circuit, the potential energy no longer takes the form of an apparent parabola (from which the harmonic spectrum derives), but instead is characterized by a cosine wave form. Therefore, the Josephson junction is a key factor in making the oscillator non-harmonic, allowing us to identify a unique, tractable quantum two-level system, as shown in Fig. 1d. At this time, the energy level difference can be expressed as:
The larger the difference between the two adjacent energy levels, the better the control. At the same time, once nonlinearity is added, the dynamics of the system is dominated by the dominant energy in the above formula, which is reflected in the ratio of EJ/EC . Over time, superconducting qubits have trended toward circuit designs with EJ ≤ EC . Because, in the opposite case of EJ ≤ EC , the qubits become highly sensitive to charge noise, which is more difficult to mitigate than flux noise, thus making it difficult to achieve high coherence; furthermore, current techniques allow for high coherence in the Hamiltonian Having more flexibility in engineering the inductive (or potential) part of the , operating at the limit of EJ ≤ EC will make the system more sensitive to potential Hamiltonian changes.
2) Quantum Hamiltonian Engineering
1. Tunable qubits
To achieve fast gate operation with high fidelity, many quantum processor architectures have tunable qubit frequencies. For example, in some cases we need to bring two qubits into resonance to exchange energy, and we also need to separate them during idling to minimize interactions.
A widely used technique today replaces a single Josephson junction with a loop interrupted by two identical "junctions" to form a DC superconducting quantum interference device (DC-SQUID), as shown in Figure 3a. Due to the interference between the two ends of the SQUID, the effective critical current of the two junctions can be reduced by applying a magnetic flux through the loop. One degree of freedom can be eliminated by exploiting this condition: the energy ( EJ ) of the Josephson junction can be adjusted by the external magnetic flux Φext (through the SQUID critical current). At this time, the EJ variation obeys a sinusoidal function, and the frequency of the qubits can be adjusted periodically by Φext , as shown in Fig. 3b.
Figure 3. Block circuit representation of capacitive shunt qubits. a. and b. are symmetric transmon qubits whose Josephson energy EJ is shunted by a capacitor, resulting in a charging energy EC ; c. and d. are asymmetric transmon qubits; e. and f. are C-shunt Flux qubits with a small junction (red) shunted to two larger junctions (orange); g. and h. are C-shunt Fluxonium qubits with a small junction inductively shunted to a large array of N-junctions.
As can be seen from Figure 3, the symmetric transmon and asymmetric transmon in the project do not change the circuit topology, but have a profound impact on practical applications. The flux sensitivity is suppressed over the entire tunable frequency range (as shown in Fig. 3d). Asymmetric transmons introduce a small frequency tuning range, sufficient to compensate for the manufacturing overhead without introducing an unnecessarily large susceptibility to magnetic flux noise, maintaining high coherence; another example, the surface-encoded scheme of adiabatic controlled phase (CPHASE) gates A specific frequency configuration between qubits is required to avoid frequency crowding problems, and asymmetric transmons are well suited for their well-defined frequency range.
Overall, as quantum processors scale up and manufacturing techniques improve, asymmetric transmons may have wider applications in the future.
2. Greater Antunity: Flux Qubits and Fluxonium
Transmon-type qubits still generate the same sinusoidal potential regardless of whether they are symmetric or not. Therefore, the characteristics and design of qubits have not fundamentally changed. In particular, the limited anharmonicity in transmon-type qubits inherently leads to significant residual excitation of high-energy states, undermining the performance of gate operations. Flux qubits (flux qubits, Fig. 3e) change the topology of the circuit, reshaping the potential energy curve. Each "knot" in Figure 3e is associated with a phase variable, and the flux condition can again eliminate one degree of freedom. Under this operation, the potential energy may take on the profile of a single well (γ ≥ 2) or a double well (γ < 2), as shown in Fig. 2f. The dual-well case is named persistent-current flux qubit (PCFQ). The most important feature of PCFQ is that its anharmonicity can be much larger than that of transmon, and it can have a longer relaxation time.
Flux qubits illustrate that the properties of qubits can be engineered by choice of circuit parameters. An extension of this idea is fluxonium qubits (Fig. 3g), in which the number of array "junctions" in flux qubits increases dramatically in some cases , even up to 100. Specifically, the potential energy can be viewed as a quadratic term modulated by a sinusoidal term. Similar to rf-SQUID-type magnetic flux qubits, the kinetic energy inductance of Josephson junction arrays is typically much larger than the geometrical inductance of the wires in rf-SQUIDs, This makes it possible to design transition matrix elements to achieve millisecond T1 coherence times.
3) Interactive Hamiltonian Engineering
So, how to achieve entanglement between individual quantum systems?
1. Physical coupling: capacitance and inductance
In superconducting circuits, the physical form of coupled energy is an electric or magnetic field (or a combination thereof). To achieve capacitive coupling, a capacitor can be placed between the two participating voltage nodes in the circuit. Figure 4a shows an example of direct capacitive coupling between two transmon-type qubits. In the case of inductive coupling, the coupling mechanism is the mutual inductance shared by the two loops, a typical example being two tightly positioned (rf-SQUID-type) magnetic flux qubits, as shown in Fig. 4c. To achieve mutual inductance, two loop circuits are brought close to each other, or even overlapped to share the inductance of a wire or Josephson junction.
Figure 4 Schematic diagram of the capacitive and inductive coupling scheme between two superconducting qubits (labeled 1 and 2). a. Direct capacitive coupling, the voltage nodes of the two qubits V1 and V2 are connected by a capacitor Cg ; b. Capacitive coupling is performed by a coupler in the form of a linear resonator; c. Direct inductive coupling, two qubits are coupled by mutual inductance M12 ; d. Inductively couple with a frequency-adjustable coupler through mutual inductance M1C and M2C .
2. Coupling axis: horizontal and vertical
Lateral coupling can be designed between qubits and harmonic oscillators, as shown in Figure 4b where two transmon-type qubits are capacitively coupled to a central resonator, a structure also known as cavity quantum electrodynamics (cQED). It has many useful applications in superconducting quantum information architecture, such as high-fidelity readout, cavity bus, quantum memory, cat-state quantum computing, etc.
Longitudinal coupling can create entanglement without energy exchange. In this case, the intermediate qubit mode can also be used as a coupler, as shown in Figure 3d, an additional rf-SQUID is used to adjust the coupling, and the coupling strength can be adjusted by the flux bias of the coupler SQUID. Tunable couplers can provide a wide range of coupling strengths and many more ways to achieve high-fidelity entanglement gates.
In addition to pure lateral, longitudinal qubit coupling, there are also hybrid types of interactions: longitudinal for qubits, but transverse for harmonic oscillators in qubit-resonator systems. Such a model is called longitudinal, but in reality it is only longitudinal to one participating system. For example, in the application of quantum annealing, the longitudinal and lateral coupling (longitudinal coupling for mapping problems and lateral coupling for improving annealing performance) enable independent control.
How to manipulate superconducting qubits to implement quantum algorithms?
1) Superconducting qubits
According to different degrees of freedom, superconducting qubits are mainly divided into three categories: charge qubits, flux qubits, and phase qubits. We distinguish these superconducting qubits mainly based on the EJ/EC ratio.
Figure 6. Superconducting qubit circuit diagram. a. Charge qubit consisting of a Josephson junction and a capacitor; b. Flux qubit. L is the loop inductance, and the energy level structure of the qubit can be adjusted by changing the bias magnetic flux Φ; c. Phase qubit. Adjusting the bias current Ib can tilt the potential energy surface.
On the basis of these three superconducting qubit prototypes, many new superconducting qubits are derived: such as Transmon-type qubits, C-type shunt magnetic flux qubits, Fluxonium, 0-π qubits, hybrid qubits, etc.
1) Transmon type qubit
Specifically, Transmon, Xmon, Gmon, 3D Transmon, etc., are currently the most popular superconducting qubits due to the simplicity and flexibility of the "cavity quantum electrodynamic" (cQED) architecture: transmon's operating system is widely used As for the current quantum computing experiment, which was developed by the Yale team around 2007, Google has also realized the "superiority of quantum computing" on the quantum computer based on this system.
Figure 7 Overview of cavity quantum electrodynamics (cQED). cQED aims to study the quantum behavior of atoms (ions) confined in specific spaces, such as micro-optical cavities, high-quality microwave cavities, confined quantum devices, etc.
Fig. 8 Schematic diagram of Transmon qubit and its circuit. a. Efficient circuit model of Transmon qubits. CB is a large capacitor in parallel with a superconducting quantum interference device (SQUID). Lr and Cr are connected in parallel to form the equivalent circuit of the readout resonator; the circuit on the far right is the magnetic flux-biased SQUID of the SQUID; b. Schematic diagram of the two-dimensional structure of the Transmon qubit.
Xmon can be seen as an improved version of Transmon, consisting of a cross-capacitor coupled to a common transmission line through a resonant cavity. Each Xmon is controlled by two independent control lines: an XY control line and a Z control line, which can be used to rotate the quantum state in the X, Y, and Z directions. Xmon qubits combine fast control, long coherence and direct connectivity for scalable superconducting quantum computing.
Figure 9 a. Optical micrograph of Xmon qubit; b. SQUID magnified image; c. Qubit circuit.
Based on Xmon qubits, Gmon uses a junction connection that acts as a tunable inductor to control the strength of the coupling. The Gmon structure can avoid the frequency crowding problem caused by fixed coupling, and its applications range from quantum computing to quantum simulation.
Figure 9 Optical micrograph of two inductively coupled Gmon qubits.
The extra inductive coupler introduces extra incoherent channels and the device layout becomes very complicated. In 2018, Yan et al. proposed a simple and general tunable coupler scheme: a general three-body system with chain geometry, and the central mode is a tunable coupler. The center mode can be constructed with any flux-tunable circuit in which the resonant frequency can be tuned.
The biggest feature of 3D Transmon is that the planar transmission line cavity is replaced by a 3D waveguide cavity. The advantages include: the cavity has a larger mode volume and is less sensitive to surface dielectric loss; secondly, the architecture provides good scalability for qubits. Control the electromagnetic environment. Therefore, this structure can suppress qubit decoherence while maintaining sufficient coupling to the control signal. However, scalability will be the main difficulty if one wants to build large-scale devices based on 3D Transmon.
Figure 10 a. Schematic diagram of a sensing qubit within a 3D cavity; b. Photograph of a half 3D aluminum waveguide cavity.
2) Three Josephson Junction (3-JJ) Magnetic Flux Qubits
Three-Josephson junction (3-JJ) flux qubits consist of a micron-sized ring with three or four Josephson junctions. The reduced loop size in the 3-JJ flux qubit results in a reduced susceptibility of the flux qubit to flux noise. In this structure, the two qubit states have persistent currents in opposite directions, and quantum superposition can be obtained by pulsing the enclosed magnetic flux with the current in the control wire.
Figure 11 Schematic diagram of 3-JJ flux qubits. Two junctions have the same Josephson coupling energy EJ and the third junction has a smaller Josephson coupling energy α EJ .
3) Capacitive shunt (C-shunt) magnetic flux qubit
The main source of decoherence is charge noise, which arises primarily from charge fluctuations in two regions separated by a smaller Josephson junction. The C-shunt magnetic flux qubit architecture introduces an extra capacitance shunted in parallel with the smaller Josephson junction in the loop. This shunt capacitor reduces the charge energy, so the effects of dominant charge noise are suppressed.
Figure 12 Schematic representation of C-shunt flux qubits. A capacitor Cs is connected in parallel to the smaller Josephson junction to reduce the charging energy associated with the b and c regions.
4) Fluxonium
Fluxonium is used to address inductance and offset charge noise. In the Fluxonium architecture, a series array of large capacitance tunneling junctions is connected in parallel with a small junction. When the system oscillation frequency is lower than the plasma frequency, the series array of large junctions can effectively behave as an inductive wire. Such a large inductance acts as a low-pass filter, so low-frequency changes in charge on small nodes are short-circuited by this large inductance, reducing the qubit's sensitivity to charge noise.
Figure 13 Circuit representation of Fluxonium qubits
In March 2022, the quantum laboratory of Alibaba Dharma Academy successfully designed and manufactured a two-bit fluxonium quantum chip, achieving a single-bit control accuracy of 99.97%, and a two-bit iSWAP gate control accuracy of up to 99.72%, among the highest in the world. The best level is an important step for the advantages of fluxonium from theory to practice.
5) 0-π qubits
The 0-π qubit adopts a symmetric circuit design to obtain interleaved double potential wells: the two ground-state wave functions of a qubit are highly localized in their respective potential wells and are not separated from each other. The transition matrix elements between the corresponding two ground state energy levels are very small, so 0-π qubits are insensitive to charge and magnetic flux noise.
Figure 14. Circuit diagram of a. 0-π superconducting qubit. The circuit has a ring with four nodes connected by a pair of Josephson junctions ( EJ,CJ ), a bulk capacitor (C) and a superconductor (L); b. In the absence of a magnetic field, The dual-well potential function V(θ,φ) of the circuit. The ground state of the 0 valley is located along θ=0, and the lowest state of the π valley is located along θ=π.
6) Plasonium qubits
Plasonium was proposed by the team of Pan Jianwei of the University of Science and Technology of China in September 2021.
In plasonium qubits, with the help of shunt inductance, the charge degree of freedom becomes a continuous variable, and the charge bias noise can be completely eliminated by gauge transformation. Therefore, a small shunt capacitor can be used to increase anharmonicity and reduce device size without being affected by charge noise. Plasonium can be viewed as a transmon with asymmetric composite junctions, including single junctions and junction arrays, that can reduce flux sensitivity. Thus, when ignoring other decoherent channels that are typically less than flux noise, the plasmanium qubits have the largest high frequency band.
Figure 15 Plasonium qubits under the electron microscope.
7) Hybrid qubits
Different quantum systems have their own advantages, and the hybrid system can combine the advantages of different quantum systems: coupling nitrogen-vacancy (NV) color centers in diamond with superconducting flux qubits. The hybrid system takes advantage of both systems: the flux qubits are controllable, but their coherence time is short, which can be used as a control element; the coherence time of NV color centers is long, which can potentially be used as a super Long-term memory for quantum processors.
Figure 16. Experimental setup of a hybrid system coupling superconducting flux qubits with electron spin ensembles in diamond. a. Diamond crystals (red box) stuck to flux quantum orbitals; b. NV color centers.
2) Superconducting qubit gates
Quantum logic can be accomplished by a set of one-qubit and two-qubit gates.
Single-qubit operations transform an arbitrary quantum state from one point on a sphere to another by rotating the Bloch vector (spin) by a certain angle around a specific axis. For example, the I gate does not rotate the state of the qubit; the X gate performs π rotations on the X axis; similarly, the Y and Z gates perform p rotations on the Y and Z axes, respectively; the S gate performs p/ 2 rotations, the T-gate performs a p/4 rotation around the Z axis. The Hadamard Gate performs a p-rotation about the diagonal of the x-axis: a p-rotation on the diagonal axis of the xz plane.
Figure 17 Single-qubit gate. The name, short description, circuit representation, matrix representation, input/output truth table, and Bloch sphere representation of each gate.
Two-qubit logic gates take two qubits as input.
Figure 18 State-of-the-art high-fidelity, two-qubit gates in superconducting qubits. a. Sorted by year of first demonstration, gate time refers to highest fidelity qubit gate; b. Full name. CZ(ad.), adiabatic control phase gate; √iSWAP, square root of iSWAP; CR, cross resonance; √bSWAP, square root of Bell–Rabi SWAP; MAP, microwave-activated phase gate; RIP, resonator-induced phase gate; CNOTL , logical CNOT gate; CNOTT-L, long-range logical CNOT gate; cF, fixed frequency, T, tunable, 3D F, fixed frequency transmon qubit in a three-dimensional cavity, BEQ, Bose-encoded qubit. For all non-boson-encoded quantum gates, the qubits are of the transmon class (except for the first demonstration of √iSWAP using phase qubits, and the first demonstration of CR using capacitive shunt qubits). Term in parentheses refers to coupling element; d. Implemented with phase qubits; e. Determined by interleaved random Clifford fiducials; f. By repeatedly applying logic gates to various input states and observing state fidelity decay as applied gates g. Determined by quantum process tomography; h. Realized with capacitively shunted flux qubits. About the shaded colors: blue represents logic gates implemented using flux-tunable qubits, pink represents gates implemented using only microwave symbols; green represents a combination of tunable and fixed frequency components; purple represents Bose-coded qubit gates .
Typically, the first qubit is the control qubit and the second is the target qubit: a unitary operator is applied to the target qubit and depends on the state of the control qubit. Common examples are controlled NOT gates (CNOT gates) and controlled phase gates (CZ or CPHASE gates). The CNOT gate puts the control bit in the state |1⟩ and flips the state of the target bit; the CPHASE gate first puts the control bit in the state |1⟩, and then applies a Z gate to the target bit; the iSWAP gate can be composed of CNOT gates and single-qubit gates Construct.
Figure 19 Two-qubit gate. Controlled Not (CNOT) gate and Controlled Phase gate (CPHASE or CZ). For each gate, the name, short description, circuit representation, matrix representation, and input/output truth table are listed.
A general set of one-qubit and two-qubit gates is sufficient to implement an arbitrary quantum logic, which means that this set of gates can in principle reach "arbitrary" states in a multi-qubit state space. How to do this efficiently depends on the choice of quantum gates that make up the gate set. At the same time, every single-qubit and two-qubit gate is reversible; that is, given the output state, the input state can be uniquely determined.
Multi-qubit gates with more than two qubits can also be implemented on superconducting quantum systems. For example, three transmon qubits are coupled to a microwave resonator to form a Toffoli gate (CCNOT controlled-controlled-not gate); CCZ (controlled-controlled-Z) gate and CCCZ ( Controlled-Controlled-Controlled-Z) Gate.
3) Using qubit gates
Capacitive coupling between resonators (or feeders) and superconducting quantum fields allows microwave control to achieve single-qubit rotation and some two-qubit gates; for flux-tunable qubits, local magnetic fields can be used to tune individual qubits Frequency of. This enables single-qubit rotation in the Z-axis as well as multiple two-qubit gates.
Every quantum computing architecture has gate operations, which are easier to implement at the hardware level than other gates (sometimes referred to as the architecture's "native gates"). In general, one wants to keep the overall number of time steps (indicating the circuit "depth") of the applied gates, and wants to use as many native gates as possible to reduce the time spent in synthesis.
1. Single-qubit gates
Capacitive coupling of microwaves to superconducting circuits can be used to drive single-qubit gates. A superconducting qubit is typically coupled to a microwave source (or "qubit drive"), as shown in Figure 20.
Figure 20 a. Schematic diagram of a typical qubit driver. A microwave source provides a high frequency signal (ω LO ) and an arbitrary waveform generator (AWG) provides a pulse envelope (s(t)), sometimes with a low frequency component ω AWG produced by the AWG . The IQ mixer combines these two signals to produce a waveform Vd(t) at a frequency of ω d = ω LO ±ω AWG that resonates with the qubit. (b) An example of how the gate sequence is transformed into the waveform produced by the AWG. Colors indicate I and Q components. (c) A pulse of X π /2 acts on a |0⟩ state and produces a |-i⟩=1/√2(|0⟩-i|1⟩) state.
2. Two-qubit gate
Single-qubit gates supplemented by entangled two-qubit gates can form a general-purpose quantum computing gate group.
Figure 21. Schematic diagram of the hierarchy of two coupled qubits (including higher levels), indicating the transitions used in the iSWAP, bSWAP, CPHASE and MAP gates.
i. iSWAP (Swap) Gate
The capacitive coupling between the qubits is turned on for a certain time (inversely proportional to the coupling strength in radial frequency), an iSWAP gate is implemented, which acts to exchange excitation between the two qubits and adds an i=eiπ /2 phase.
Figure 22 iSWAP gate implementation circuit diagram
ii. CPHASE (controlled phase) gate
In the iSWAP gate, we assume that the high energy levels of superconducting qubits do not play a role. However, it turns out that for the case of transmon-like bits (with negative anharmonicity), higher energy levels can be used to directly generate CPHASE gates.
CPHASE gates are applied to capacitively coupled transmon superconducting qubits. When two bits are in the excited state |11⟩, a phase needs to be imposed on them ( -1=ei π); considering that coupling |01⟩↔|10⟩ generates iSWAP gates, we expect to avoid the Energy level crossings such as |11⟩↔|20⟩ and |11⟩↔|02⟩. A flux-tunable implementation of the CPHASE gate relies on avoiding this higher-level crossover.
Figure 23 Equivalent circuit using CPHASE gate and native single-qubit gate.
iii. CR gate
A common potential disadvantage of iSWAP and CPHASE gates is that their operation requires flux-tunable qubits. The need for flux tuning increases the susceptibility of the device to flux noise, resulting in improved decontamination rates. To this end, cross-resonance (CR) gates have been developed to operate fixed-frequency superconducting qubits, typically with longer lifetimes and lower susceptibility to magnetic flux noise.
Figure 24. Circuit schematic of two fixed-frequency transmon qubits coupled through a resonator. Qubit 1 driven at the frequency of qubit 2 leads to the emergence of a CR gate.
Dispersive readout allows researchers to map the qubit's degrees of freedom onto the classical response of a linear resonator, translating the readout optimization process into obtaining the best signal-to-noise ratio (SNR) for the microwave signal used to probe the resonator. Quantum measurements can be described as the entanglement of qubit degrees of freedom with a "pointer variable" of a measurement probe with a quantum Hamiltonian, followed by a classical measurement of the probe.
Figure 25 a. Experimental setup for dispersive qubit reading. The probe tone of the resonator is generated, shaped and timed with an arbitrary waveform generator (AWG) and fed into the cryostat. The reflected signal S11 is first amplified in a parametric amplifier, then in a low-noise HEMT amplifier, down-converted by heterodyne mixing, and finally sampled in a digitizer; b. When the qubit is in the ground state |0⟩ (blue) and excited state |1⟩ (red), reflection amplitude | S11 | and phase θ response of a resonator with linewidth κ at frequency 2χ/2π ; c. Corresponding complex plane representation. The highest state discrimination is obtained when probing the resonator between two resonances, the dashed line in (b).
In qubit control, measurement equipment, or the surrounding local environment, random, uncontrollable physical processes are sources of decoherence noise that reduce qubit operational fidelity. How to solve this problem in engineering?
1) Noise type
There are many sources of noise affecting quantum systems, which can be divided into two main types: systematic noise and random noise.
1. System noise
Systematic noise arises from processes that can fix control or readout errors. For example, applying microwave pulses to a qubit is expected to bring about a 180-degree rotation; however, a poorly tuned control field causes the pulse to only slightly over- or under-spin the qubit by a fixed amount. This fundamental error is "systematic" in that it results in the same rotation error every time it is applied.
Once a system error is identified, it can be corrected with proper calibration or hardware improvements.
2. Random noise
Random noise is caused by random fluctuations in the coupling parameters of the qubits. For example, thermal noise of a 50Ω resistor in a qubit control wire will have fluctuations in voltage and current, known as Johnson noise; or, an oscillator that provides a carrier for qubit control pulses may have amplitude or phase fluctuations; in addition, quantum Randomly fluctuating electric and magnetic fields in the bit environment (metal surface, substrate surface, metal) can couple to qubits. This creates unknown and uncontrollable fluctuations in one or more qubit parameters, leading to qubit decoherence.
3. Noise intensity and qubit sensitivity
The degree to which a qubit is affected by noise is related to the amount of noise and the susceptibility of the qubit to that noise. The former is usually a materials science and manufacturing problem: whether scientists can make devices with lower noise levels; the latter is a qubit design problem.
Consequently, materials science, manufacturing engineering, electronic design, cryogenic engineering, and qubit design are all striving to create devices with high coherence. In general, scientists should work to eliminate sources of noise and then design qubits that are insensitive to residual noise.
2) Decoherence model
1. Bloch sphere representation
A qubit's response to noise depends on how the noise is coupled to it: longitudinally or laterally. Using the qubit quantization axis as a reference, this can be illustrated using a Bloch sphere diagram.
Figure 5. Representation of lateral and longitudinal noise in a Bloch sphere. a. Bloch sphere representation of the quantum state | φ ⟩=α|0⟩+β|1⟩; b. The longitudinal relaxation comes from the energy exchange between the qubit and the environment, due to the transverse noise coupled to the quantum in the xy plane bits, and drives the transition |0⟩↔|1⟩. A qubit in state |1⟩ emits energy to the environment and relaxes to |0⟩ at a rate of Γ1 ↓ (blue arced arrow). Likewise, a qubit in state |0⟩ absorbs energy from the environment and excites to |1⟩ at a rate of Γ1 ↑ (orange arced arrow); c. Pure dephasing arises from longitudinal noise along the z-axis, which makes quantum The bit frequency fluctuates randomly and the Bloch vector along the x-axis will spread clockwise or counterclockwise around the equator, depolarizing at the rate of Γφ ; d. Transverse relaxation leads to loss of coherence due to energy relaxation and depolarization coherent combination.
A Bloch sphere is a unit sphere used to represent the state of a qubit. If the Bloch sphere is imagined as the earth, then the north pole represents the ground state |0⟩ and the south pole represents the excited state |1⟩. The z-axis connects the North and South poles and is called the "vertical axis"; similarly the xy plane is the "horizontal plane" with "horizontal axes" x and y. Figure 4a shows a Brock sphere representing a “Brock vector” of states |φ⟩=α|0⟩+β|1⟩ ( |α|2+|β|2=1 ), where the Bloch vector is the unit Length, which can connect the center to any point on the surface of the sphere: the surface of the unit sphere represents the pure state, and its interior represents the mixed state.
2. Decoherence model
1) Longitudinal relaxation
The longitudinal relaxation rate Γ1 describes the depolarization along the quantized axis and is often referred to as "energy decay" or "energy relaxation". At this point, the polarization qubit is completely in the ground state |0⟩ at the "North Pole", in the excited state |1⟩ at the South Pole, and in a completely depolarized mixed state at the center of the Bloch sphere.
As shown in Fig. 4b, longitudinal relaxation is caused by "transverse noise" passing through the x- or y-axis, and depolarization is caused by energy exchange with the environment, which generally results in an "upconversion rate" Γ1 ↑ (excitation from |0⟩ to |1⟩) and the “downconversion rate” Γ1 ↓ (relaxing from |1⟩ to |0⟩), which together constitute the longitudinal relaxation rate Γ1 : Γ1=1/T1=Γ1↑+Γ1↓ , T1 is the characteristic time scale (the shortest time span that can reflect the essential changes of the system).
2) Pure dephasing
The pure dephasing rate Γφ describes the depolarization in the x–y plane of the Bloch sphere, which distinguishes it from other dephasing processes (eg, energetic excitation, relaxation). As shown in Fig. 4c, pure dephasing is caused by "longitudinal noise" coupled to the qubit through the z-axis, which causes the Bloch vector to push forward or backward in the rotating frame.
There are several important differences between pure dephasing and energy relaxation. First, in contrast to energy relaxation, pure dephasing is not a resonance phenomenon: any frequency noise can change the frequency of a qubit and cause dephasing; second, pure dephasing is different from spontaneous energy relaxation, which is elastic (without exchange of energy with the environment) and in principle "reversible": quantum information can be preserved.
3) Transverse relaxation rate
transverse relaxation rate Γ2=1/T2=Γ1/2+Γφ describes the loss of coherence of the superposition state, as shown in Fig. 4d.
3. Common noise types
1) Charge noise
Charge noise is ubiquitous in solid-state devices. Charged undulators from interfacial dielectric defects or charge traps, junction tunneling barriers, and the substrate itself. For example, for transmon qubits, the electric field between the capacitive plates traverses and couples to the dielectric defects on the surface of the metal plates (for side-plate type capacitors) or the dielectric between the capacitive plates (for parallel-plate type capacitors). This means that the charge noise is mainly responsible for the longitudinal relaxation rate Γ1 and is generally modeled as a combination of inverse frequency noise and Nyquist noise, also known as "Ohmic" noise.
2) Magnetic flux noise
Another frequently observed noise in solid state devices is magnetic flux noise. This noise arises from random flips of spins (magnetic dipoles) left on the surface of the superconducting metal that make up the qubits, resulting in random fluctuations in the effective magnetic field, biasing the qubits with tunable magnetic flux.
Although much is known about the defect statistics and numbers of putative flux noises, their precise physical manifestations remain uncertain. Recent studies suggest that adsorbed oxygen molecules may be responsible for the flux noise.
3) The number of photons fluctuates
In circuit QED architectures, the photon number fluctuation of the resonator is another major source of decoherence. The fluctuations originate from residual photons in the resonator, usually caused by radiation from the higher temperature stages in the dilution refrigerator.
4) Quasiparticles
"Quasiparticles," or unpaired electrons, are another important source of noise in superconducting devices. Quasiparticles can cause T1 relaxation and pure dephase Tφ, depending on the qubit type, the bias point, and the location of the "junction" where the tunneling event occurs.
4. Engineering noise mitigation
What are the current technical directions for reducing noise, or reducing its impact on decoherence (sensitivity)?
1) Material improvement
Many efforts have been made to reduce noise defects due to materials and manufacturing: for charge noise, efforts to reduce the number of defects, such as substrate cleaning, substrate annealing; for flux noise, flux defects are being characterized experimentally The behavior and properties of quasiparticles; for residual quasiparticles, it has been demonstrated that incorporating quasiparticle traps in circuit designs can reduce quasiparticle populations, such as in classical digital logic or operating in the presence of thermal radiation.
2) Design improvements
The susceptibility of qubits to noise can be reduced by design. It has been demonstrated that changing the geometry of the capacitor to increase the electric field mode volume can reduce the electric field density in thin dielectric regions that cause losses. This effectively reduces the participation of defects, making the qubits less sensitive to noise sources. In another example, split transmons built using asymmetric "junctions" are less sensitive to flux noise than symmetrical "junctions", but at the cost of reduced frequency tunability.
3) Dynamic error suppression
The spin-echo technique, which destroys free evolution by p-pulsing, is extremely effective in mitigating pure dephasing by refocusing the coherent phase dispersion caused by low-frequency noise. More advanced versions, such as the CPMG-sequence, use multiple p-pulses to interrupt the system more frequently, pushing the filter band to higher frequencies, a technique known as "dynamic decoupling".
4) Cryogenic engineering
In the case of photon emission noise, in addition to applying dynamic decoupling techniques, there are studies aimed at reducing the thermal photon flux reaching the device. These include optimizing the attenuation of cryogenic devices, adding absorptive "black" materials to absorb stray thermal photons, and adding additional cavity filters for thermalization.
1) Quantum Annealing
Superconducting qubits form the basis of a quantum annealing platform. The operation of quantum annealing is to find a ground state for a given Hamiltonian (usually the classical Ising Hamiltonian) and associate this state with a solution to an optimization problem.
2) Cavity-based quantum information processing
Parallel to planar superconducting qubits is the development of three-dimensional cavity-based superconducting qubits.
In this system, quantum information is encoded in a superposition of coherent photonic modes in the cavity. Due to the high quality of the three-dimensional cavity, the cat states can be highly coherent. This approach has small hardware overhead for logical qubit encoding; and observable errors due to single-photon loss in the cavity make it suitable for implementing asymmetric error-correcting codes.
3) Cryogenic technology and software development
Although dilution refrigerators are ready-made commercial products, how to optimize signal routing and fast data processing details in a scalable way also needs to be developed.
In terms of control software, several commercial and free software packages currently exist for interfacing with quantum hardware, such as QCoDeS, the related pyCQED, qKIT, and Labber. There are also a large number of quantum circuit simulation and compilation packages in development, such as Qiskit, Forest (with pyQUIL413), ProjectQ, Cirq, OpenFermion, and Microsoft QDK, which provide higher-level programming languages to compile and/or optimize quantum algorithms.
4) Quantum Error Correction
While qubit lifetimes and gate fidelity have greatly improved over the past few decades, error correction is still required to reach large-scale processors.
While certain strategies exist to scale physical qubits, for truly large-scale algorithms to solve practical problems, quantum data must be embedded with error correction schemes. Surface-code quantum error correction schemes have now been demonstrated in superconducting qubits, however, proving that logical qubits have longer lifetimes than fundamental physical qubits remains an outstanding challenge.
Although surface codes are promising quantum error-correcting codes due to their relatively relaxed fault-tolerance thresholds, they cannot implement universal gate groups in a fault-tolerant manner. This means that error correction gates in surface codes need to be supplemented, for example, with T gates, to be universal. Currently, such gates can be implemented through a technique known as "magic state extraction," but demonstrating the extraction and injection of surface-encoded logic states remains an open challenge.
5) The superiority of quantum computing
A big challenge for superconducting qubits in the coming years: demonstrating the superiority of quantum computing. The basic idea is to use qubits and algorithm gates to demonstrate a computation that is beyond the reach of classical computers (assuming some reasonable computational complexity conjectures).
references:
[1] JP Dowling and GJ Milburn, “Quantum technology: The second quantum revolution,” Philos. Trans. R. Soc. London, A 361, 1655–1674 (2003).
[2]MA Nielsen and IL Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition,
