Tencent has made the latest progress in quantum circuit compilation
Superconducting qubit is a relatively mature implementation scheme of quantum computing [1,2]. At present, hundreds of superconducting qubits have been integrated in the experiment, and the bit coherence time can be as long as hundreds of microseconds [3]. However, there are still many problems to be solved in the experiment, among which ZZ crosstalk is one of the biggest factors limiting the fidelity of quantum gate / circuit [4,5]. ZZ crosstalk is a kind of additional interaction caused by the coupling between superconducting qubits (as shown in Figure 1). At present, the mainstream scheme to suppress this kind of crosstalk in the experiment is to integrate a hardware coupler [6] between bits to realize the controllable switch of coupling between bits. However, this will complicate the design and manufacture of quantum chips, increase the processing difficulty, and aggravate the possibility of decoherence.
Figure 1 Schematic diagram of ZZ crosstalk between superconducting qubits
Fortunately, the ZZ crosstalk intensity is fixed and can be eliminated by optimizing the control waveform (hereinafter referred to as the waveform) in principle. However, it is not scalable to directly and simply optimize the waveform of large-scale quantum circuits - the time and memory required by the optimization algorithm increase exponentially with the number of quantum bits in the circuit [7].
To solve this problem, we propose a scalable anti noise software hardware joint compilation optimization framework. The framework suppresses the ZZ crosstalk around the quantum gate by optimizing the waveform of each quantum gate (hardware level), and makes full use of this ability through the scheduling of quantum gates in the circuit (software level optimization), so as to realize the crosstalk suppression at the whole circuit level. The advantages of this scheme are: 1 It will not increase the complexity of quantum chip design and processing; 2. Both hardware and software layer optimization algorithms are effective and scalable, and are suitable for large-scale quantum computing. The simulation results show that the computational fidelity of 4-12 qubit circuits can be improved by up to 81 times (11 times on average). Through a set of Ramsey experiments, we verify that waveform optimization can largely eliminate the influence of all ZZ crosstalk around the quantum gate. This idea of software and hardware joint compilation also provides an enlightening idea for more general noise control in large-scale quantum computing.
Overview of hardware and software joint compilation framework
Superconducting quantum computing is realized by quantum circuits running on superconducting quantum chips. Usually, circuits contain many single qubit gates and double qubit gates. Only for single and double qubit gates, we can quickly find a way to optimize the waveform to eliminate the influence of ZZ crosstalk. Describe the form of implementing H(t)=HCrtl + λHXtalk , as shown in Figure 2; Wherein HCrtl(t) is determined by the waveform applied to the corresponding bit, HXtalk represents the normalized ZZ crosstalk term, λ describes the intensity of ZZ crosstalk. When the surrounding qubits are idle (the white circle in figure. 2), we can eliminate the influence of ZZ crosstalk between the specified area (the gray part in Figure 2) and other parts by optimizing HCrtl(t). This optimization is usually fast and effective at the single and double qubit gate level, and constructs the basic module of hardware level optimization.
Figure 2 : (a) Single bit gate and (b) two bit gate: the black (white) circle is the qubit with (without) control waveform. The gray area represents the basic area for waveform optimization. The red arrow indicates the cross region boundary ZZ crosstalk.
When applying the optimized waveform of anti ZZ crosstalk to some bits, the surrounding bits must be in a free idle state in order to achieve the suppression effect. Therefore, when considering the whole circuit, we need to consider that under this constraint, rearrange the quantum gate [8], insert the auxiliary identity gate for suppressing crosstalk, and try our best to compress the depth of the whole circuit while maximizing the suppression of ZZ crosstalk, so as to reduce the influence of decoherence in the calculation operation.
Based on this idea, we propose a framework for software and hardware joint compilation of quantum circuits, as shown in Figure 3. Specifically: 1 We first optimize the waveforms of several general quantum gates (including auxiliary identity gates) that need to be used in the circuit to suppress all ZZ crosstalk around them; 2. Given the specific circuit, we rearrange the circuit according to the needs of restraining ZZ crosstalk, and insert some auxiliary identity quantum gates appropriately. 3. Convert these quantum gates into the waveform optimized in step 1 according to their space-time position, and then send the waveform of the whole circuit to the quantum chip for operation. Next, we will introduce this framework.
Figure 3 Joint compilation optimization framework of waveform control and quantum gate scheduling
Suppression of cross region crosstalk through waveform optimization
We take the single qubit gate in Fig. 2 as an example to introduce the waveform optimization scheme. The idea of two bit gate is similar. Suppose a unitary transformation U1 is applied to a bit. Unlike traditional waveform optimization, which only cares about the fidelity of the quantum gate U1, we also need to consider reducing the influence of ZZ crosstalk on the quantum bits around the U1. Therefore, we pay more attention to the generalized fidelity a of the quantum gate picture acting on the quantum bit F(U(T),U1UINbr(a)), where 
is the unitary transformation actually experienced by a and its surrounding bits, and t is the length of the whole waveform. Then the whole waveform optimization problem can be reduced to the following problems:
Here 
is the quantum gate waveform generating 
, 
is the unitary transformation generated by 
. We consider two kinds of single bit quantum gates: 1 
rotating about the X axis; 2. Auxiliary identity gate 
for noise suppression only. For these two gates, we consider four single qubit waveforms respectively: 1 Gaussian reference waveform; 2. Waveform after global optimization (optctrl); 3. The waveform obtained by pertubation based (PERT) of perturbation expansion term based on λ; 4. Dynamic corrected gate (DCG). Optctrl and pert both use gradient method to optimize specific objective functions. DCG uses a combination of Gaussian waveforms to complete the quantum gate. The DCG of 
and identity gate is composed of 5 and 2 Gaussian waveforms respectively.
First, consider two bits: Q1-Q2. The waveform acts on Q1, and only the influence of ZZ crosstalk is considered. The fidelity of generalized single bit gate under several waveforms is shown in Figure 4 - in the case of 
(weak crosstalk), the three waveform optimization methods of 2, 3 and 4 can improve the generalized fidelity by several orders of magnitude and greatly reduce the influence of all ZZ crosstalk around. Pert is especially effective because it is optimized for ZZ crosstalk, while other methods have a wider range of applications.
Figure 4 Relationship between non generalized fidelity and crosstalk intensity λ under different control waveforms
For two qubit gates, we try to optimize a specific gate 
. Considering that the four qubits Q1-Q2-Q3-Q4 are arranged in a straight line, 
acts on Q2-Q3. After waveform optimization, the generalized fidelity is shown in Figure 5. Similar to the single bit case, optctrl and pert control schemes can also significantly reduce the impact of ZZ crosstalk on Q1 and Q4 in a small time.
Figure 5 Generalized non fidelity of 
(a) different waveform control of 
;(b) 
with different strength is controlled by pert.
To sum up, using the gradient based scheme and DCG scheme, we can easily obtain a good waveform to suppress ZZ crosstalk.
Through the scheduling of quantum gates and waveform optimization
With the crosstalk suppression capability at the hardware level, we can explore the quantum gate in the scheduling circuit and properly insert the auxiliary identity quantum gate to maximize the suppression of all ZZ crosstalk in the whole circuit and minimize the sacrifice of parallelism. We call this scheduling scheme zzxsched.
Let's look at a simplified case, that is, given a layer of quantum gates that can run simultaneously. Take the circuit in Fig. 6 (a) as an example (Fig. 6 (b) is the corresponding topology diagram) to reflect the main idea of scheduling. If the optimized waveform is directly adopted, the ZZ suppression effect is shown in Figure 6 (b): the dotted line part is the suppressed crosstalk edge, and the red edge represents the non suppressed crosstalk - there are 13 non suppressed edges at this time.
Figure 6 (a) Single layer circuit example (b) direct scheduling + waveform optimization. The dotted line is the side where ZZ crosstalk has been suppressed.
Figure 7 Two schemes of suppressing ZZ crosstalk by inserting auxiliary identity gate in single-layer circuit
Further, more edges can be eliminated by inserting an optimized auxiliary identity gate. Figure 7 shows two schemes: plan a inserts identity gates on bits 1 and 11, reducing four red edges; Further: Plan B inserts additional identity gates on 3 and 13, reducing two red edges. Therefore, the problem is: how to insert the identity gate to minimize the number of red edges (crosstalk) in a given layer of circuit. This problem is related to the maximum cut of the graph. It is a NP complete problem and cannot be solved effectively in general. However, note that superconducting qubits are planar graphs, and the maximum cut problem of planar graphs has polynomial algorithm. Based on this, we propose an optimization algorithm, optimal suppression, to optimize the position of inserting identity gates.
Let 
be the number of nodes (qubits), edges and vertices of the dual graph corresponding to the superconducting chip, 
, then the complexity of this algorithm is 
. compared with the exponential time algorithm for the general maximum cut problem, this polynomial complexity algorithm is much more efficient, so that it can be extended to large-scale quantum circuits.
In addition, the execution time points of different quantum gates in a layer circuit can be changed to disassemble a layer circuit into multiple layers. We still take Figure 6 (a) as an illustration. If we disassemble the circuit into two layers (as shown in Fig. 8), the first layer can insert auxiliary identity gates in bits 1, 4, 11 and 14, leaving only three unchecked red edges; The second layer inserts auxiliary identity gates in bits 1, 3, 5, 7, 11, 13 and 15, and the ZZ crosstalk of this layer is completely suppressed (without red edge). In this way, only 3 edges of the layered circuit are not suppressed, while there are 6 edges of the non layered circuit.
Figure 8 After a single-layer circuit is decomposed into two-layer circuits, an auxiliary identity gate is inserted to suppress ZZ crosstalk
Through this example, we find that layered quantum gate scheduling can make the arrangement of quantum gates in each layer more decentralized, so it is more suitable to insert auxiliary identity gates to suppress more ZZ crosstalk. Therefore, we propose a heuristic algorithm to repeatedly split the executable circuit of each layer, and use the optimal suppression algorithm to optimize the insertion position of the auxiliary identity gate. The calculation complexity of this algorithm is only xschemal suppress polynomial.
Experimental and simulation evaluation results
In order to evaluate the effect of software hardware joint compilation framework, we test it in experiment (waveform optimization) and simulation (overall circuit suppression effect).
Firstly, we verify the principle at the hardware level: we need to ensure that all cross-border ZZ crosstalk can be suppressed by the control waveform on the qubit at the same time. Ramesy experiment is usually used to calibrate the phase decoherence time T2 of qubits, and can also be used to measure the offset of bit frequency. Therefore, we use a set of Ramsey experiments to characterize the intensity of ZZ crosstalk. We consider three transmon superconducting qubits arranged in a line q1-q2-q3. We conducted three groups of Ramsey experiments on Q2 to investigate the effects of Q1 and Q3 on Q2, marked as a, B and C. Fig. 9, FIG. 10 and FIG. 11 are experimental diagrams respectively. Experiment a was the conventional reference group; In experiments B and C, we are right: 1 Q2 (Experiment B) apply DCG constant waveform; 2. Apply DCG identical waveform to Q1 and Q3 (Experiment C); In this way, the influence of Q1-Q2 and Q2-Q3 pair ZZ crosstalk on Q2 is investigated.
Figure 9 Experiment a: Ramsey benchmark experiment for Q2, Q1 and Q3 are in 
, or 
.
Figure 10 Experiment B: Ramsey experiment after applying DCG identical waveform to Q2, Q1 and Q3 are in the 
or 
.
Figure 11 Experiment C: Ramsey experiment on Q2; Q1 and Q3 apply DCG constant waveform at the same time
Figure 12 Comparison of the results of experiments a and B: (a) Q1 and Q3 are in the 
(Orange Line) or 
(blue line); (b) Q1, Q3 in 
(Orange Line) 
(blue line)
Figure 13 In experiment (a) (above), Q1 and Q3 are in 
(Orange Line) and 
(blue line); Experiment (b) (lower left), Q1 and Q3 are in the 
; Experiment (c) (lower right).
We find that the crosstalk intensity of the two pairs of ZZ is reduced by two orders of magnitude respectively. From Fig. 13 (bottom left), we find that the identity waveform acting on Q2 can simultaneously suppress the corresponding two groups of cross region ZZ crosstalk. From Figure 13 (lower right), we find that the identical waveform acting on Q1 and Q3 at the same time can also greatly suppress the influence of two groups of ZZ crosstalk on Q2. It can be seen that by optimizing the waveform, we can greatly suppress the influence of cross region ZZ crosstalk at the same time - which is the physical basis of our compilation framework.
On this basis, we use full quantum simulation to evaluate the circuit fidelity of the compilation framework on the actual quantum algorithm circuit, and assume that the ZZ crosstalk intensity obeys the Gaussian distribution of 
.
First, we consider the influence of only ZZ crosstalk in the circuit. We select 6 types of commonly used circuits (with a scale of 4-12 qubits and running in 3 × On the superconducting quantum chip of 3), Gaussian controlled wave mode and maximum parallel scheduling (gau + parsched) are used as evaluation benchmarks. Considering zzxsched + optctrl and PERT, the simulation results are shown in Figure 14. It can be seen that the overall circuit fidelity has been greatly improved (about 81 times at most). More interestingly, the more the number of qubits, the greater the fidelity of the test circuit. When the system is expanded to only 12 bits, the fidelity of the original method is reduced to 20%, which is completely impractical. But after using our method, it can generally be increased to more than 80% or 90%. Therefore, we believe that the crosstalk suppression effect of software hardware joint compilation scheme is better for the larger scale quantum computing circuit.
Figure 14 Fidelity evaluation of quantum circuits with different waveform + scheduling schemes
In addition to the influence of ZZ crosstalk, qubits also endure the decoherence process, which is usually characterized by the decoherence times T1 and T2. Assuming T1 = T2 to facilitate simulation, the circuits of six quantum characteristics are simulated. The relevant results are shown in FIG. 15 - when T1 and T2 are greater than 100US, the effect of decoherence on the effectiveness of our scheme is limited, and the overall circuit fidelity can still be improved by up to 10 times, which is basically consistent with the results in FIG. 14.
Figure 15 Fidelity evaluation of quantum bit circuits -- the influence of decoherence on different waveform + scheduling schemes
Zzxsched usually controls the increase of execution time to less than 2 times (FIG. 16), which can greatly suppress ZZ noise at the expense of limited parallelism. This trade-off is worth it. As we have seen in figures 14 and 15, the overall circuit fidelity has still been greatly improved.
Figure 16 Maximum parallel compiler (parsched) and anti ZZ crosstalk compiler (zzxsched) generate circuit execution time comparison
Summary
We propose a quantum circuit compilation framework based on the combination of software and hardware according to the characteristics of ZZ crosstalk. At the microwave control level of a single quantum gate, we suppress the propagation of ZZ crosstalk by optimizing the waveform of the quantum gate. At the software level, the reasonable arrangement, scheduling and transformation of quantum gates are used to suppress the influence of ZZ crosstalk in the whole calculation process. Different from many noise suppression methods for nisq, the computational complexity of this software hardware joint compilation framework can be proved to be effective, so it can be applied to large-scale quantum circuits. It is observed that the larger the circuit size is, the better the improvement effect is.
In the future, we will integrate the software and hardware joint compilation framework into the real quantum computing working environment on the basis of the completed work, and complete the experimental principle verification. We hope to maximize the computing potential of quantum chips in the nisq era and promote the development of quantum algorithms at a low cost. We also hope that this idea of software hardware joint compilation can help us suppress the impact of more general noise.
Link:
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