We propose an efficient O(N 2 )-parameter ansatz that consists of a sequence of exponential operators, each of which is a unitary variant of Neuscamman's cluster Jastrow operator. The ansatz can also be derived as a decomposition of T2 amplitudes of the unitary coupled cluster with generalized singles and doubles, which gives a near full-CI energy, and reproduces it by extending the exponential operator sequence. Because the cluster Jastrow operators are expressed by a product of number operators and the derived Pauli operator products, namely the Jordan-Wigner strings, are all commutative, it does not require the Trotter approximation to implement to a quantum circuit and should be a good candidate for the variational quantum eigensolver algorithm by a near-term quantum computer. The accuracy of the ansatz was examined for dissociation of a nitrogen dimer, and compared with other existing O(N 2 )-parameter ansatzs. Not only the original ansatzs defined in the second-quantization form but also their Trotterized variants, in which the cluster amplitudes are optimized to minimize the energy obtained with a few, typically single, Trotter steps, were examined by quantum circuit simulators.

Background and Objectives: Leg extension angle is important for increasing the propulsion force during gait and is a meaningful indicator for evaluating gait quality in stroke patients. Although leg extension angle during late stance might potentially also affect lower limb kinematics during the swing phase, the relationship between these two remains unclear. This study aimed to investigate the relationship between leg extension angle and knee flexion angle during pre-swing and swing phase in post-stroke gait. Materials and Methods: Twenty-nine stroke patients walked along a 16 m walkway at a self-selected speed. Tilt angles and acceleration of pelvis and paretic lower limb segments were measured using inertial measurement units. Leg extension angle, consisting of a line connecting the hip joint with the ankle joint, hip and knee angles, and increments of velocity during pre-swing and swing phase were calculated. Correlation analysis was conducted to examine the relationships between these parameters. Partial correlation analysis adjusted by the Fugl-Meyer assessment-lower limb (FMA-LL) was also performed. Results: On the paretic side, leg extension angle was positively correlated with knee flexion angle during the swing phase (r = 0.721, p < 0.001) and knee flexion angle and increments of velocity during the pre-swing phase (r = 0.740–0.846, p < 0.001). Partial correlation analysis adjusted by the FMA-LL showed significant correlation between leg extension angle and knee flexion angle during the swing phase (r = 0.602, p = 0.001) and knee flexion angle and increments of velocity during the pre-swing phase (r = 0.655–0.886, p < 0.001). Conclusions: Leg extension angle affected kinematics during the swing phase in post-stroke gait regardless of the severity of paralysis, and was similar during the pre-swing phase. These results would guide the development of effective gait training programs that enable a safe and efficient gait for stroke patients.

Many stroke patients rely on cane or ankle-foot orthosis during gait rehabilitation. The purpose of this study was to investigate the immediate effect of functional electrical stimulation (FES) to the gluteus medius (GMed) and tibialis anterior (TA) on gait performance in stroke patients, including those who needed assistive devices. Fourteen stroke patients were enrolled in this study (mean poststroke duration: 194.9 ± 189.6   d ; mean age: 72.8 ± 10.7   y ). Participants walked 14 m at a comfortable velocity with and without FES to the GMed and TA. After an adaptation period, lower-limb motion was measured using magnetic inertial measurement units attached to the pelvis and the lower limb of the affected side. Motion range of angle of the affected thigh and shank segments in the sagittal plane, motion range of the affected hip and knee extension-flexion angle, step time, and stride time were calculated from inertial measurement units during the middle ten walking strides. Gait velocity, cadence, and stride length were also calculated. These gait indicators, both with and without FES, were compared. Gait velocity was significantly faster with FES ( p = 0.035 ). Similarly, stride length and motion range of the shank of the affected side were significantly greater with FES (stride length: p = 0.018 ; motion range of the shank: p = 0.02 6). Meanwhile, cadence showed no significant difference ( p = 0.238 ) in gait with or without FES. Similarly, range of motion of the affected hip joint, knee joint, and thigh did not differ significantly depending on FES condition ( p = 0.115 ‐ 0.529 ). FES to the GMed and TA during gait produced an improvement in gait velocity, stride length, and motion range of the shank. Our results will allow therapists to use FES on stroke patients with varying conditions.

In stroke patients, the impact of lower limb physical functions on the leg extension angle remains unclear. We set out to reveal the physical impairments of the affected side in such patients that were associated with leg extension angle during gait. Twenty-six stroke patients walked for 16 m at a spontaneous speed. During walking, the leg extension angle and the increment of velocity during late stance, as an indicator of propulsion, were measured by inertial measurement units. The Berg balance scale (BBS), Fugl-Meyer assessment-lower limb, and motricity index-lower limb (MI-LL) were also evaluated. Stepwise multiple regression analysis was employed to reveal functions associated with the leg extension angle on the affected side. A path analysis was also used to confirm the relationship between the extracted factors, leg extension angle, and gait speed. Multiple regression analysis showed that the BBS was significantly related to the leg extension angle on the affected side (p < 0.001). Path analysis revealed that the leg extension angle was also indirectly affected by the MI-LL and that it affected gait speed via propulsion on the affected side. These findings could guide the prescription of effective gait training for improving gait performance during stroke rehabilitation.

Unilateral knee extension restriction might change trunk alignment and increase mechanical load on the lumbar region during walking. We aimed to clarify lumbar region mechanical load during walking with restricted knee extension using a musculoskeletal model simulation. Seventeen healthy adult males were enrolled in this study. Participants walked 10 m at a comfortable velocity with and without restricted right knee extension of 15° and 30° using a knee brace. L4–5 joint moment, joint reaction force, and muscle forces around the lumbar region during walking were calculated for each condition. Peaks of kinetic data were compared among three gait conditions during 0%–30% and 50%–80% of the right gait cycle. Lumbar extension moment at early stance of the bilateral lower limbs was significantly increased in the 30° restricted condition ( p ≤ 0.021 ). Muscle force of the multifidus showed peaks at stance phase of the contralateral side during walking, and the erector spinae showed force peaks at early stance of the bilateral lower limb. Muscle force of the multifidus and erector spinae increased with increasing degree of knee flexion ( p ≤ 0.010 ), with a large effect size (η2 = 0.273–0.486). The joint force acting on L4–5 showed two peaks at early stance of the bilateral lower limbs during the walking cycle. The anterior and vertical joint force on L4–5 increased by 14.2%–36.5% and 10.0%–23.0% in walking with restricted knee extension, respectively ( p ≤ 0.010 ), with a large effect size (η2 = 0.149–0.425). Restricted knee joint extension changed trunk alignment and increased the muscle force and the vertical and anterior joint force on the L4–5 joint during walking; this tendency became more obvious with increased restriction angle. Our results provide important information for therapists engaged in the rehabilitation of patients with knee contracture.

Bridging exercise is commonly used to increase the strength of the hip extensor and trunk muscles in physical therapy practice. However, the effect of lower limb positioning on the joint and muscle forces during the bridging exercise has not been analyzed. The purpose of this study was to use a musculoskeletal model simulation to examine joint and muscle forces during bridging at three different knee joint angle positions. Fifteen healthy young males (average age: 23.5 ± 2.2 years) participated in this study. Muscle and joint forces of the lumbar spine and hip joint during the bridging exercise were estimated at knee flexion angles of 60°, 90°, and 120° utilizing motion capture data. The lumbar joint force and erector spinae muscle force decreased significantly as the angle of the knee joint increased. The resultant joint forces were 200.0 ± 23.2% of body weight (%BW), 174.6 ± 18.6% BW, and 150.5 ± 15.8% BW at 60°, 90°, and 120° knee flexion angles, respectively. On the other hand, the hip joint force, muscle force of the gluteus maxims, and adductor magnus tended to increase as the angle of the knee joint increased. The resultant joint forces were 274.4 ± 63.7% BW, 303.9 ± 85.8% BW, and 341.1 ± 85.7% BW at a knee flexion angle of 60°, 90°, and 120°, respectively. The muscle force of the biceps femoris decreased significantly with increased knee flexion during the bridging exercise. In conclusion, the knee flexion position during bridging exercise has different effects on the joint and muscle forces around the hip joint and lumbar spine. These findings would help clinicians prescribe an effective bridging exercise that includes optimal lower limb positioning for patients who require training of back and hip extensor muscles.