prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What may happen if the VR headset lenses are exposed to sunlight or strong light?

Read the following text and answer briefly.

Context 1: '用户指南 * User Guide 02 CN 11 EN * 包装内含 使用前注意事项 快速引导 产品部件详情说明 操作说明 02 02 03 06 08 01 
•本产品支持在系统设置中进行瞳距调节 , 调节时请务必注意，最小瞳距可能会碰触鼻梁。当您佩戴头盔后，您 “显示”中进行手动调节，请注意设置使用不合适的瞳距，可能会引起视觉重影或者眼睛疲劳。 可在“设置” ► •本产品“护眼模式”经德国 TÜV Rheinland 低蓝光认证，通过软件算法降低三色通道中的蓝光量，来达到保护 “护眼” “色彩调节” 眼睛的作用，该模式下画面颜色偏黄，您可根据个人喜好在“设置” 中激活或关闭此功能。 ““显示” ► ► ► 包装内含： VR 头盔 / 手柄 × 2 / 1 5V AA 碱性干电池 × 4/ 眼镜支架 / 遮光鼻托 / 手柄挂绳 × 2 / USB-C 电源适配器 / USB-Cto C 2 0 数据线 / 快速指南 / 用户指南 / 安全与质保指南使用前注意事项 •本产品在开阔的室內环境使用体验最佳，建议至少预留 2×2 米 的空间。使用前请确认身体没有不适且周围环 境安全，特别是佩戴头盔在室内行走移动时，要尽量避免发生意外。 •不建议 12 岁及以下儿童使用本产品，建议将头盔、手柄和配件置于儿童够不到的位置，13 岁以上青少年须在 成人监护下使用，以免发生意外。 •本产品无近视调节功能，近视用户请佩戴眼镜使用并尽量避免近视眼镜被头盔的光学镜片磨伤或刮伤。建议在 使用和收纳时注意防护光学镜片，避免尖锐物体划伤镜片，擦拭清洁时请使用柔软的眼镜布，否则可能划伤镜片， 影响视觉效果。 •长时间使用可能引发轻微的昡晕或者眼疲劳，建议使用 30 分钟后适当休息，可通过眼保健操或观看远处物体缓 解眼疲劳。如果您的身体感到任何不适，请立即停止使用。如果不适持续，请咨询医生。 •当头盔镜片被阳光或紫外线照射时（尤其在户外、阳台、窗台及汽车内存放时），可能导致屏幕出现永久性黄斑。 请尽量避免该情况发生，此种屏幕损坏不在产品的质保范围内。 *本产品最终外观及功能以实物为准，部分地区包装内含物品有所差异，本说明仅供参考。 02 CN
六自由度 VR 体验 本产品可以追踪头盔和手柄前、后、左、右、上、下和旋转的运动状态，您在现实中的肢体运动会实时反映在虚 拟世界中。 由于没有任何线缆的束缚，您在虚拟世界自由探索时请确保游玩区域的安全。 1  建议准备一个整洁安全的体验空间：至少 2×2 米；保持房间明亮，避免在只有单色的墙或大面积玻璃、镜子类 反射物以及许多移动画面和物体的空间中使用。 2  撕下 VR 头盔前端摄像头上的保护膜，并佩戴手柄挂绳。 3  根据开机后的画面提示进行游玩区域的设定。 ❶ 安装电池 按箭头方向拔出电池盖侧边的绝缘纸 快速引导 提示：本产品虚拟的安全区提醒功能，不能完全保证您在设定好的游戏区域中的安全，请时刻注意周围的安全情况。 提示：建议使用 1 5V AA 碱性电池。 按照图示拨动电池盖拨钮打开电池盖更换电池。 03 CN
❷ 手柄开机 ❸ 头盔开机 ❹ 佩戴头盔，调节至清晰舒适的位置 首次开机：拔出绝缘纸，手柄自动开机（蓝灯闪烁） 非首次开机：短按手柄 Home 键开机（蓝灯闪烁） 长按头盔电源键 2 秒（蓝灯常亮） 调节旋钮转动绑带，使后脑垫套在头上，微调绑带长度及佩戴位置至视野清晰 04 提示：近视用户请佩戴眼镜或镜片插件使用，本产品不具备近视调节功能。 CN
❺ 微调顶绑带 微调顶绑带使其受力以减少额头压力 ❻ 瞳 距 调 节 在系统设置：“设置” ► “显示”界面中进行瞳距调节，点击“+”或“-”按钮可微调瞳距直至画面清晰 64mm 请勿 强行 掰动镜 筒，以 免造 成损坏 ！ 请注 意设 置使用 不合适 的瞳 距，可 能 会引起 视 觉重影 或 者眼睛 疲 劳。准 确 的瞳距 设 置有助 于 获得清 晰 的图像 并 减少眼睛 疲劳。 05 CN
产品部件详情说明 头盔状态指示灯 蓝灯常亮：开机进行中或工作状态 黄灯常亮：充电中，电量低于 98% 红灯常亮：充电中，电量低于 20% 绿灯常亮：充电完毕，电量大于 98% 或 充满 蓝灯闪烁：关机进行中 红灯闪烁：电量低于 20% 指示灯熄灭：休眠或关机 06 ① 电源键 开机：长按 2 秒 关机：长按 5 秒 复位：长按 10 秒 开机时，短按休眠 ② ③ ④ ⑤ 状态指示灯 贴脸泡棉 音量键 彩色透视摄像头 使用时请勿遮挡 ⑥ ⑦ ⑧ 顶部绑带 可拆卸 绑带旋钮 环境追踪摄像头 使用时请勿遮挡 ⑨ ⑩ ⑪ USB-C 接口 左 / 右喇叭 距离传感器 佩戴头盔后，系统自动唤醒 摘下头盔后，系统自动休眠 ⑫ ⑬ 眼球追踪摄像头 此功能仅 Pro 版支持 使用时请勿遮挡 面部追踪摄像头 此功能仅 Pro 版支持 使用时请勿遮挡 CN
手柄状态指示灯 熄灭：已连接或者关机 蓝灯常亮：固件升级模式 蓝灯闪烁：连接中 红蓝灯交替慢速闪烁：等待配对 ① ② 摇杆 菜单键 ③ Home 键 开机 : 短按关机 : 长按 6 秒退出应用 : 短按屏幕中心校正 : 长按 1 秒④ ⑤ ⑥ ⑦ 状态指示灯 抓握键 截屏键 扳机键 ⑧ ⑨ 电池盒 打开：拨动拨钮，电池盒弹出 安装：按压直至自动锁紧 追踪光环 使用时请勿遮挡 注：手柄挂绳可按图示将粗绳穿过细绳并锁紧在手柄尾端 07 CN
手柄硬件复位 如果手柄出现按 Home 键和任何按键均无反应或者头盔中虚拟手柄卡死不动的问题可拆装电池重新启动手柄。 近视用户配戴 本设备不具备近视调节功能，头盔可支持佩戴镜框宽度小于 150mm 的大多数标准眼镜。 操作说明 头控模式 未连接手柄的情况下，您可通过转动头部光标及点击头盔音量加减按键进行操作。 切换主控手柄射线 在主控菜单下，短按对应手柄的扳机键可以切换主控手柄的射线。 屏幕中心校正 戴着头盔直视前方，按住手柄 Home 键（或头控模式下头盔上的音量减键）1 秒以上，进行屏幕中心的校正将菜 单拉到当前视野朝向位置。 断开手柄 长按手柄 Home 键直至手柄状态指示灯红灯亮起并伴随振动产生时即可松手，此时手柄关机并断开与头盔的连接。 您无需刻意进行手柄关机操作，在以下状态下手柄会自动关机省电： •头盔进入深度休眠时（摘下头盔后一段时间） •头盔手柄管理界面解绑手柄时 •头盔关机时 添加新手柄 如需添加新手柄（头盔最多可同时连接一对手柄，即左右手柄各一只），或解绑手柄后再次连接 , 可进入“设置” “手 柄”，点击“配对”，同时按住手柄 Home 键和扳机键直至手柄状态指示灯红蓝交替闪烁时即可松开，然后根据 头盔画面提示操作。 ► 休眠 / 唤醒 方式一：摘下头盔一段时间后，系统自动休眠；戴上头盔时，系统自动唤醒。 方式二：短按头盔电源键也可以进行休眠或唤醒操作。 硬件复位 头盔硬件复位 如果头盔出现短按头盔电源键没有反应或头盔的画面卡死等问题，可以长按头盔电源键 10 秒以上重新启动头盔。 08 CN
安装眼镜支架 安装遮光鼻托 如您存在眼镜摩擦光学镜片或者压迫鼻梁的问题，请按照图示安装眼镜支架以增加间隔空间。 您可根据佩戴的舒适度选择是否安装。 如您感觉鼻子处漏光影响体验，请按照图示安装遮光鼻托配件。 由于眼睛空间密闭可能加剧起雾及出汗问题，您可根据喜好选择是否安装。 ❶ 摘下贴脸泡棉 ❷ 将眼镜支架按照图示安装在产品上 ❸ 将贴脸泡棉按照图示安装眼镜支架上 ❶ 摘下贴脸泡棉 ❸ 安装贴脸泡棉❷ 将遮光鼻托按照图示方式安装在贴脸泡棉上 注：按照图示拆卸眼镜支架 09 CN
更换贴脸泡棉 贴脸泡棉多次清洁和长时间使用后会变色和质地变软，您可酌情更换新泡棉。 更换顶绑带 摘下贴脸泡棉 ❸ 安装贴脸泡棉 按照图示捏住顶绑带金属扣，往下压到底然后抽出 ❷ •购买优质热门应用 •畅 聊 社 区， 与 众 多 PICO 玩 家 一起探索 VR 世界 •管理设备更便捷 •参与丰富互动活动 •更多精彩内容等你来发现 ❶ 微 信公 众 号：PICO VR抖音：PICO官 方 旗 舰 店哔 哩 哔 哩：PICO-VR官 方微 博：PICO-VR ❶ ❷ 10 CN
In The Box: VR Headset / 2 Controllers / 4 1 5V AA Alkaline Batteries / Glasses Spacer / Nose Pad / 2 Controller Lan- yards / USB-C Power Adapter / USB-C to C 2 0 Data Cable / Quick Guide / User Guide / Safety and WarrantyGuide Important Health & Safety Notes • This product is designed and intended to be used in an open and safe indoor area, free of anytripping or slipping hazards  To avoid accidents, remain conscious to the potential confines ofyour physical area and respect the boundary of your virtual area whenever you see it  Be sure towear the lanyards when using the Controllers  Make sure that there is enough space around yourhead and body (at least 2 meters by 2 meters) to stretch your arms to avoid damage or injury toyourself, others, and your surroundings  • This product is not recommended for children aged 12 and under  It is recommended to keep headsets,controllers and accessories out of the reach of children  Teenagers aged 13 and over must use it underadult supervision to avoid accidents  • This product is designed to accommodate most prescription glasses  Make sure to wear the VR Headsetin a manner in which the VR Headset lenses do not rub or impair your prescription lenses  • Prolonged use may cause dizziness or eye fatigue  It is recommended to take a break every 30 minutes Try relieving your eyestrain by looking at distant objects  If you feel any discomfort, stop using the prod- uct immediately  If the discomfort persists, seek medical advice • Do not expose the optical lenses to direct sunlight or other strong light sources  Exposure to directsunlight may cause permanent yellow spot damage on the screen  Screen damage caused by sunlightexposure or other strong sources of light is not covered by the warranty  • This product supports interpupillary distance (IPD) adjustment in system settings  When adjusting,please be aware that with the minimum IPD, it may touch the bridge of the nose  You can adjust the IPDaccording to your actual interpupillary distance in "Settings"►"Display"  Please note that using inap- propriate IPD may increase the risk of discomfort  • This product has an “Eye Protection Mode”, certified by TÜV Rheinland (Germany), which can protectyour eyes by reducing blue light in the three color channels using software algorithms  The screen ap- pears yellowish in this mode and you can turn this feature on/off in "Settings"►"Display"►"Color"►“- Eye Protection”  • Protect optical lenses during use and storage to prevent damage, such as scratches or exposure tostrong light or direct sunlight  * Product and packaging are updated regularly, and the functions and contents of the standalone headset may be upgraded in the future Therefore, the content, appearance and functionality listed in this manual and product packaging are subject to change and may notreflect the final product  These instructions are for reference only  * Carefully read this user guide before using the product and share this information with any other users, as it contains important safetyinformation  Keep the user guide as reference for the future  11 EN
6 Degrees of Freedom VR The device can track your translational and rotational movements in all directions (up/down, left/right,forward/backward, pitch, roll, and yaw)  Your movements in the real world will be captured and translatedto what you see in the virtual world when using the appropriate content  Ensure a safe environment before you start your VR experience  1  Clear a safe indoor area of at least 2 meters by 2 meters  Keep the room bright, avoid spaces with main- ly single-colored walls, glass, mirrors, moving pictures or other similar objects  2  Remove the protective film that covers the headset front cameras  Wear the lanyards connected to theControllers  3  Set up your environment by following instructions on the VR Headset screen  Install Batteries ❶Pull the tab to remove the insulating paper  Quick Guide 2 m 2m This product can not guarantee your safety with guardian system, you will need to always pay attention to the surrounding safety  * Note: 1 5V AA alkaline batteries should be used Slide the toggle according to arrow direction toopen the battery case  12 EN
Power on the Controller ❷ First Start: The Controller will start automaticallyafter removing the insulating paper  Others: Short press the Home button for 1second until the status indicator flashes blue Power on the VR Headset ❸ Long press the Power button for 2 seconds untilthe status indicator turns blue Wear Your Headset for a Comfortable Fit and View ❹ Adjust the strap dial to turn the strap so that the back of your head rests on the padding  Fine-tune thelength and position of the strap to give a clear view  * Note: You can use this product with prescription glasses or lenses insert  13 EN
Fine-tune the Top Strap ❺ Fine-tune the head strap to reduce pressure on the forehead  Interpupillary Distance (IPD) Adjustment ❻ In System Setting, go to “Setting” ► “Display” to adjust IPD, tap “+” or “-” button to slightly adjust IPDuntil the picture is clear  14 64mm Please note that inappropriate IPD setting may cause ghosting or eyestrain Accurate IPD setting helps you get a clear imaging and ease eyestrain  EN
Product Details VR Headset Status Indicator Legend Blue: Powered on with battery over 20% Yellow: Charging: Battery is less than 98% Red: Charging: Battery is less than 20% Green: Charging: Battery is more than 98% or charge complete Blue flashing: Shutting down Red flashing: Battery is less than 20% Off: Sleeping or Powered off Power Power on: Long press for 2 seconds Power off: Long press for 5 seconds Hardware reset: Long press for 10 seconds Short press to enter sleep or wake up Status Indicator Face Cushion Volume ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ RGB See Through Camera Do not block during use  Top Strap Removable Strap Dial Tracking Cameras Do not block during use  ⑨ ⑩ ⑪ USB-C Interface Left/Right Speaker Proximity Sensor The system wakes upwhen the VR headset isput on, sleeps when VRheadset is taken off  ⑫ ⑬ Eye Tracking Cameras Pro version only  Do not block during use  Face Tracking Camera Pro version only  Do not block during use  15 EN
Controller Status Indicator Legend Off: Connected or Powered off Blue: Firmware updating in progress Blue flashing: Searching for connection Red and blue flashing alternately: Pairing in progress 16 Joystick Menu ③ ① ② Home Power on: Short pressPower off: Long press for 6 secondsReturn home screen: Short pressScreen recentering: Press for 1 secondStatus Indicator Grip Capture Trigger ④ ⑤ ⑥ ⑦ ⑧ ⑨ Battery Case Open: Slide down the toggle andpop up the battery case  Lock: Push the battery case to lock  Tracking Ring Do not block during use  * Note: Pass the Controller Lanyardthrough the string as shown andlock at the end of the Controller EN
Operating Instructions Headset Control Mode If the Controller is not connected, you can interact with the home screen by moving your head to directthe crosshairs over your intended selection and clicking the Volume Up/Down button on the VR Headset  Switch the pointer of the master Controller In the home screen, short press the Trigger of the corresponding Controller to switch the pointer of themaster Controller  Screen re-centering Wear the VR Headset and look straight ahead, press and hold the Home button of the Controller or VRHeadset ( or the Volume Down button of the VR Headset in head control mode) for more than 1 second tore-center the screen  Disconnect the Controller Press and hold the Home button until the status indicator turns red and the Controller vibrates Controllers will automatically shut down to save power in the following cases:When the VR Headset enters deep sleep (a while after the VR Headset is taken off)When the Controller is unpairedWhen the VR Headset is powered off Add a new Controller If you need to add a new Controller (the VR Headset can only connect one left Controller and one rightController) or reconnect with an unpaired Controller  Go to “Settings” ► “Controller”, click on “Pair” Press and hold the Home button and the Trigger of the Controller at the same time until the red and bluelights of the Controller flashing alternately, and then follow the instructions on the VR Headset screen  Sleep / Wake up Option 1 (Proximity Sensor) Take off VR Headset for automatic sleeping: wear the VR Headset for automat- ic waking up  Option 2 (POWER Button) Press the Power button of the VR Headset for manual sleeping or waking up  Hardware reset VR Headset reset If the visual in the VR Headset freezes, or the VR Headset does not respond after short press the Powerbutton, you can press the Power button of the VR Headset for more than 10 seconds to reboot the VRHeadset  Controller reset If the virtual Controller, the Home button or any buttons of the Controller doesn\'t respond, remove andreinstall the battery case to restart the Controller  The VR Headset Adjustment This device has no myopia adjustment function  The VR Headset allows wearing most standard glasseswith a frame width of less than 150mm  to install Glasses Spacer to increase the space  You can install or not according to your situation  17 EN
Install Glasses Spacer Install Nose Pad If you have glasses collision with headset lens or pressure on the bridge of nose, please follow the pictureto install Glasses Spacer to increase the space  You can install or not according to your situation  If you feel light leaking from your nose, please follow the picture to install Nose Pad to block the light You can consider having it installed at your own discretion  Disassemble the Face Cushion  Install the Glasses Spacer on the Headset  ❸ ❶ ❷ Install the Face Cushion on the Glasses Spacer  Disassemble the Face Cushion  Install the Nose Pad on the Face Cushion  ❶ ❷ Install the Face Cushion on the Headset  ❸ * Note: Disassemble the Glasses Spacer 18 EN
Replace Face Cushion The Face Cushion will have the following phenomena such as color change, surface fluff, soft texture afterlong-term use and repeated cleaning  You can replace a new Face Cushion as needed  Replace Top Strap ❶ ❷ Disassemble the Face Cushion  Pinch the metal buckle of the top strap asshown, press it down and pull it out Install the Face Cushion on  ❸ ❷ ❶ • Purchase high-quality and trending apps • Join PICO Community and explore the VR worldwith other PICO players• Manage your device with ease • Engage in diverse and interactive activities • More exciting features waiting for you 19 EN
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ground_truth: Exposure to sunlight or strong light may cause permanent yellow spot damage on the screen.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What size chains were used in the benchmarking?

Read the following text and answer briefly.

Context 2: Paper Info

Title: Compressed quantum error mitigation
Publish Date: 10 May 2023
Author List: Maurits Tepaske (from Physikalisches Institut, Universität Bonn), David Luitz (from Physikalisches Institut, Universität Bonn)

Figure

FIG 3 The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and time t, for the infinite temperature initial state, for a denoised second-order Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 We consider evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise with p = 0 01 
FIG  4  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 0046 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is evident that the denoiser recovers all the noiseless eigenvalues from the noisy circuit 
FIG  2  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 036 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is clear that the denoiser recovers with high accuracy the noiseless eigenvalues from the noisy circuit 
FIG  3  The half-chain channel entanglement entropy S at different two-qubit depolarizing noise strengths p, for a secondorder Trotter supercircuit with Mtrot = 16 and t = 2, for a M = 4 denoiser The Trotter circuit is for a Heisenberg model with PBC of size L = 6 The different curves correspond to the different supercircuits, i e  the noisy supercircuit, the denoiser, the corresponding denoised supercircuit, and the noiseless variant 
FIG  4  The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and stacked time t, for the infinite temperature initial state, for a denoised secondorder Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 It is optimized at t = 2 and stacked up to ten times The calculations are for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarization with p = 0 01 The denoiser is affected by the same noise 
FIG 6 The distribution of the ZZ angle α of M = 2 denoisers (top panels) and M = 8 denoisers (bottom panels), with the lightest color corresponding to the denoiser for the Trotter supercircuit with t = 0 5, and the darkest color with t = 5 As usual, we consider the Heisenberg model on a periodic chain, and second-order Trotter supercircuits with depths Mtrot = 8, 16, 32, 64, which together with the denoiser is affected by a two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG  7  The sampling overhead γ of the optimized denoisers from Fig  2 of the main text, with denoiser depths M = 1, 2, 4, 6, 8 and Trotter depths Mtrot = 8, 16, 32, 64 at times t = 0 5, 1,    , 5, for the Heisenberg model on a chain with PBC affected by two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG 8 The domain wall magnetization Z dw after evolving a periodic density wall |dw |dw * with the denoised second-order Trotter supercircuits D C from Fig 2of the main text These supercircuits have various Trotter depths Mtrot = 8, 16, 32, 64, denoiser depths M = 1, 2, 4, 6, 8, and evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise of strength p = 0 01 The denoiser is affected by the same noise The non-denoised results are labelled with M = 0 and the noiseless results with p = 0 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively We see that the denoiser allows us to recover the noiseless behavior 

abstract

We introduce a quantum error mitigation technique based on probabilistic error cancellation to eliminate errors which have accumulated during the application of a quantum circuit  Our approach is based on applying an optimal "denoiser" after the action of a noisy circuit and can be performed with an arbitrary number of extra gates 
The denoiser is given by an ensemble of circuits distributed with a quasiprobability distribution  For a simple noise model, we show that efficient, local denoisers can be found, and we demonstrate their effectiveness for the digital quantum simulation of the time evolution of simple spin chains  Introduction 
-Quantum information processing has been theoretically shown to hold great promises, and quantum algorithms were developed which can in principle achieve an exponential speed-up over their classical counterparts, both for general purpose computing and quantum simulation   However, present day quantum computing prototypes still suffer from significant noise processes which hinder the execution of many potentially groundbreaking quantum algorithms  
Nontrivial quantum algorithms typically require large sequences of quantum gates, each of which introduces dissipation and hence an overall loss of coherence, eventually rendering the results useless  Until quantum error correction becomes practical, quantum error mitigation seems to be more feasible to increase the accuracy of expectation values 
Here the goal is to induce the (partial) cancellation of errors that stem from noisy quantum gates by extending the circuit corresponding to the desired algorithm with an ensemble of gates , sampled from a quasiprobability distribution  The traditional way to accomplish this is with the gatewise method from , where noise is mitigated by inverting the noise channel of each gate separately, i e  the cancellation of errors is performed for each gate on its own 
Here the local noise channel is approximated in a way such that it can be easily inverted analytically, e g  using Pauli twirling   Gates are then sampled from the inverted noise channel by interpreting it as a quasiprobability distribution  Because in this gate-wise approach every noisy gate has to be modified separately, the sign problem is exponentially large in the number of gates, limiting the practicality of the mitigation 
The success of the gate-wise approach resulted in a large body of work concerning these methods , including extensions for simultaneous mitigation of multiple gates by Pauli-twirling entire layers or variationally constructing a mitigating matrix product operator   In principle, errors during the execution of a circuit can propagate and accumulate 
These propagated errors * david luitz@uni-bonn de ≈ C

C

FIG  1  An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain  The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue) 
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser  can potentially blow up and lead to large errors for the circuit as a whole   Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit 
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution  We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel 
Essentially, the denoiser inverts a global noise channel  Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation  Method  -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise 
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm  In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate  In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps  
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise   These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise 
This supercircuit encodes an ensemble of circuits   For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters  The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values 
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible  Here C is the target unitary circuit  Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C 
We illustrate the mitigation procedure in Fig  , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer  This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser 
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits  Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it  The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values 
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth  We take this into account by imposing the same noise on the denoiser 
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count  To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor 
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel  For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i   For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond 
The resulting gate has 7 real parameters φ i   For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i  
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving  Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix 
We construct the denoiser from the noisy channels Gi = N G i   With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i e  consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer)  For reference, a general channel has 544M parameters 
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space  To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample   In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ 
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i   For quasiprobability distributions, i e  with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2  The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0 5, 1,    , 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state 
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64  In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions  All gates are affected by two-qubit depolarizing noise with p = 0 01 
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0  where sgn(η g ) is the sign of the sampled coefficient of the gth channel  γ = 1 means that all signs are positive  Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate  Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms  The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω)  
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples  We observed that decompositions with γ > 1 are crucial for an accurate denoiser  Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser 
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable  This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps 
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf  Fig  )  This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in   For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising  We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from   Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel 
Results  -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i  A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step  
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise  The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0 01 unless stated otherwise 
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0 5, 1,    , 5, and denoisers D with depths M = 1, 2, 4, 6, 8  In the top panels of Fig  we show (4) for a chain of size L = 8 as a function of time t  Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t 
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0 5, where the Trotter gates are closer to the identity than in the other cases  At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2  In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity 
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t  In the bottom panels of Fig  we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain 
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult  At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent  As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig  we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2  Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise  In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking 
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole  In Fig  we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right) 
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center  We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle 
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit  In the Supplementary Material we show the spectra for a p = 0 036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs    There we also investigate the channel entropy of the various supercircuits  
Conclusion  -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel  The required number of mitigation gates, i e  the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation 
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers  This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels 
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e g  a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone  To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions  
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks 
Nonetheless, we can use e g  matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states 
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size  Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig  , because finite-size effects appear when the light-cone width becomes comparable with system size 
1  The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p  We compare the values obtained with and without a denoiser, i e  M > 0 and M = 0, to the noiseless values (p = 0) 
The denoiser is affected by the same noise as the Trotter circuit  We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14  * david luitz@uni-bonn de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1 
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser  It is interesting to analyze the spectra of the supercircuits considered in this work 
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle  The comparison to this case is therefore instructive  In the main text, we showed an example of the spectra in Fig  for moderate noise strength  Here, we show additional data for stronger noise p = 0 036 in Fig  for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1 
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise  To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly  If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case 
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in   By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation  The right panel of Fig  shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars) 
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue  The complexity of an operator can be quantified by its operator entanglement entropy   Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits 
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i e  ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization  Then we have S = −Tr [ψ C ln ψ C ]  This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment  
In Fig  we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]  The Trotter circuit is for a Heisenberg model with L = 6 and PBC  We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S 
Here we investigate how denoised supercircuits perform upon repeated application  We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t  Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig  we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site 
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0 01 depolarizing two-qubit noise  The noisy results correspond to M = 0 and the noiseless results to p = 0  In Fig  we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times 
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig  ) as a function of M   FIG    The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times 
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8  The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0 01, which also affects the denoiser  The non-denoised results are labelled with M = 0, and the noiseless results with p = 0 
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively  The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text 
For completeness, we here present the α angles of the optimized denoisers  The results are shown in Fig  , which contains histograms for the channel count N G versus α  The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0 5 and the darkest at t = 5  The top four panels are for a denoiser with M = 2 and the bottom four with M = 8 
We consider M trot = 8, 16, 32, 64  We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity  Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8 
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities  Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text 
In Fig  we show the sampling overhead of the denoisers from Fig  of the main text  We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ  This could be related to the difference in α distributions from Fig   
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases  Here C(t) is the Trotter supercircuit for time t  In Fig  we show Z dw for the circuits from Fig 
Context 3: Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels 



ground_truth: L = 8 and L = 14.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What does the paper aim to solve?

Read the following text and answer briefly.

Context 3:  In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.