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FIG. 5: Eigenstates distribution p, after the proce- dure (simulations vs. ideal Born rule). The Born rule predicts that p, = |(E,|Wo)|?.. The model un- der consideration is the Hs Hamiltonian in Eq. (21) with the initial state in Eq. (22). The horizontal axis n labels the index of eigenstates with eigenen- ergies (E£,) ordered from the lowest to the highest. At € {100,100/3, 100/37, 100/3%, 100/3*, 100/3°} and @ is chosen randomly each time in [0,27), and for each At value we iterate 5 times. Each run is ter- minated when ((Ah)?) < 10719 and if more itera- tions are needed when all values in the At list are used, we recycle the At list from the beginning. The statistics were obtaining by averaging over 10,000 runs. The final distribution obtained from the simulations is {0.5557, 0.0733, 0.2617, 0.0077, 0.1016}.

Figure 5 Eigenstates distribution p, after the proce- dure (simulations vs. ideal Born rule). The Born rule predicts that p, = |(E,|Wo)|?.. The model un- der consideration is the Hs Hamiltonian in Eq. (21) with the initial state in Eq. (22). The horizontal axis n labels the index of eigenstates with eigenen- ergies (E£,) ordered from the lowest to the highest. At € {100,100/3, 100/37, 100/3%, 100/3*, 100/3°} and @ is chosen randomly each time in [0,27), and for each At value we iterate 5 times. Each run is ter- minated when ((Ah)?) < 10719 and if more itera- tions are needed when all values in the At list are used, we recycle the At list from the beginning. The statistics were obtaining by averaging over 10,000 runs. The final distribution obtained from the simulations is {0.5557, 0.0733, 0.2617, 0.0077, 0.1016}.

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FIG. 1: Basic picture of our algorithm. (a) The primi- tive: one ancilla is used as the control qubit for the con- trol unitary, which is jointly applied to the ancilla and the system, cU = |0)(0| @ J + |1)(1| @ e~*“"”, followed by a measurement on the ancilla in the X = o® basis. (b) Summary of the action on the input system state: |!) ~ e@™|w). This leads to a random walk picture for the algorithm.
FIG. 1: Basic picture of our algorithm. (a) The primi- tive: one ancilla is used as the control qubit for the con- trol unitary, which is jointly applied to the ancilla and the system, cU = |0)(0| @ J + |1)(1| @ e~*“"”, followed by a measurement on the ancilla in the X = o® basis. (b) Summary of the action on the input system state: |!) ~ e@™|w). This leads to a random walk picture for the algorithm.
FIG. 2: The iteration procedure using various choices of At (arbitrary unit) and ¢. See the main text for detailed discussions of the four types of choices (I) to (IV). In (a) the energy is in an arbitrary unit and the values are recorded in the iteration; in (b) the energy variances (also in an arbitary unit) are recorded.
FIG. 2: The iteration procedure using various choices of At (arbitrary unit) and ¢. See the main text for detailed discussions of the four types of choices (I) to (IV). In (a) the energy is in an arbitrary unit and the values are recorded in the iteration; in (b) the energy variances (also in an arbitary unit) are recorded.
FIG. 3: The iteration procedure using various choices of At (arbitrary unit) and ¢ similar to those in Fig. 2, except that a = /3/2 and 6 = 1/2e"®. See the main text for detailed discussions of the four types of choices (I) to (IV). In (a) the energy values (arbitrary unit) are recorded in the iteration; in (b) the energy variances (also in arbitrary units) are recorded.
FIG. 3: The iteration procedure using various choices of At (arbitrary unit) and ¢ similar to those in Fig. 2, except that a = /3/2 and 6 = 1/2e"®. See the main text for detailed discussions of the four types of choices (I) to (IV). In (a) the energy values (arbitrary unit) are recorded in the iteration; in (b) the energy variances (also in arbitrary units) are recorded.
FIG. 4: The iteration procedure using varying At and random ¢. This corresponds to the choice (IV) in the main text. Here, At is chosen from the set {100, 100/3, 100/37, 100/3°, 100/34, 100/3°}, but per- turbed by 1% of random fluctuations: At > At(1 + 0.01a) with x € [0,1). There are four different runs but with the same initial state of the system. In (a) the en- ergy values are recorded in the iteration; in (b) the energy variances are recorded. Both the energy and its variance are displayed in arbitary units.
FIG. 4: The iteration procedure using varying At and random ¢. This corresponds to the choice (IV) in the main text. Here, At is chosen from the set {100, 100/3, 100/37, 100/3°, 100/34, 100/3°}, but per- turbed by 1% of random fluctuations: At > At(1 + 0.01a) with x € [0,1). There are four different runs but with the same initial state of the system. In (a) the en- ergy values are recorded in the iteration; in (b) the energy variances are recorded. Both the energy and its variance are displayed in arbitary units.
FIG. 6: Histograms of the number of iterations to reach an eigenstate with an accuracy ((Ah)?) < 107“
FIG. 6: Histograms of the number of iterations to reach an eigenstate with an accuracy ((Ah)?) < 107“
FIG. 7: Example simulations on spectral projection for 5-qubit transverse field Ising model. (a) and (b) show the traces of energy and its variance in arbitrary units, respectively. There are three different runs (but with the same initial state of the system). Each run is terminated when ((Ah)?) < 10~'°. The phase parameter ¢ in the ancilla state is chosen randomly at each step and At is chosen from the set {100, 100/3, 100/37, 100/3°, 100/3*, 100/3°} and each At repeated 5 times. The iterations continue by recycling the At set until the desired precision is met. (c) The bottom panel compares the distribution of projected eigenstates in 10,000 simulation runs with the ideal Born rule.
FIG. 7: Example simulations on spectral projection for 5-qubit transverse field Ising model. (a) and (b) show the traces of energy and its variance in arbitrary units, respectively. There are three different runs (but with the same initial state of the system). Each run is terminated when ((Ah)?) < 10~'°. The phase parameter ¢ in the ancilla state is chosen randomly at each step and At is chosen from the set {100, 100/3, 100/37, 100/3°, 100/3*, 100/3°} and each At repeated 5 times. The iterations continue by recycling the At set until the desired precision is met. (c) The bottom panel compares the distribution of projected eigenstates in 10,000 simulation runs with the ideal Born rule.
FIG. 8: Energy variance (in an arbitrary unit) in pres- ence of decoherence. We apply the depolarizing channel at each step of the iteration, with two different « = 1073 and 10-®. Two runs are performed for each respective e. The Hamiltonian is the transverse-field Ising model with g = 0.5. It is seen that the energy variance is larger than 5e. There are 204 steps in each run.
FIG. 8: Energy variance (in an arbitrary unit) in pres- ence of decoherence. We apply the depolarizing channel at each step of the iteration, with two different « = 1073 and 10-®. Two runs are performed for each respective e. The Hamiltonian is the transverse-field Ising model with g = 0.5. It is seen that the energy variance is larger than 5e. There are 204 steps in each run.
FIG. 9: Energy (top) and energy variance (bottom), similar to the simulations in Fig. 8, except that « = 0.01 and the depolarizing channel applies only every 30 steps, starting at step 1 and ending at step 181. There are 204 steps in each run of the three runs. Both the energy and its variance are displayed in arbitary units.
FIG. 9: Energy (top) and energy variance (bottom), similar to the simulations in Fig. 8, except that « = 0.01 and the depolarizing channel applies only every 30 steps, starting at step 1 and ending at step 181. There are 204 steps in each run of the three runs. Both the energy and its variance are displayed in arbitary units.
FIG. 10: Application of our spectral projection algo- rithm in the quantum annealing as the subroutine. The figures display the energy (in an arbitrary unit) after pro- jecting to the eigenstates vs. g for the transverse-field XzY model at r = 0.5, ie. Axyzy(g = 0,r = 0.5) for (a) N = 5 qubits, and (b) N = 6 qubits. The curves represent eigenenergies as a function of g. The proce- dure starts with two different initial states: (1) [(blue) dots that start on the lowest curve] the ground state |00000) of Hxzy(g = 0,r = 0.5), and (2) [(red) dots that start on the top curve] the highest-energy state |11111) of Hx,y(g = 0,r = 0.5). All the energy levels of Hx,y(g,r = 0.5) are also shown by solid curves. (a) Due to energy level crossings, the ground state transits to a higher excited state after the crossing, and the high- est energy state transits to a lower energy state after an associated crossing. Quantum annealing does not work if there is any level crossing. (b) Due the existence of a respective small gap, the initial ground state ends up at the final ground state and the initial highest-energy state ends up also at the final highest-energy state.
FIG. 10: Application of our spectral projection algo- rithm in the quantum annealing as the subroutine. The figures display the energy (in an arbitrary unit) after pro- jecting to the eigenstates vs. g for the transverse-field XzY model at r = 0.5, ie. Axyzy(g = 0,r = 0.5) for (a) N = 5 qubits, and (b) N = 6 qubits. The curves represent eigenenergies as a function of g. The proce- dure starts with two different initial states: (1) [(blue) dots that start on the lowest curve] the ground state |00000) of Hxzy(g = 0,r = 0.5), and (2) [(red) dots that start on the top curve] the highest-energy state |11111) of Hx,y(g = 0,r = 0.5). All the energy levels of Hx,y(g,r = 0.5) are also shown by solid curves. (a) Due to energy level crossings, the ground state transits to a higher excited state after the crossing, and the high- est energy state transits to a lower energy state after an associated crossing. Quantum annealing does not work if there is any level crossing. (b) Due the existence of a respective small gap, the initial ground state ends up at the final ground state and the initial highest-energy state ends up also at the final highest-energy state.
FIG. 11: Application of our spectral projection algo- rithm in the quantum annealing as the subroutine for the transverse-field Ising model Hypr(g). The figure shows the energy (in an arbitrary unit) after projecting to the eigenstates vs. g for the 5-qubit (top panel (a)) and 6-qubit (bottom panel (b)) transverse-field Ising model. The procedure starts with two different initial states: (1) [(blue) dots that start on the lowest curve] the ground state |00000) of Hrrr(g = 0), and (2) [(red) dots that start on the top curve] the highest-energy state |11111) of Ayri(g = 0).
FIG. 11: Application of our spectral projection algo- rithm in the quantum annealing as the subroutine for the transverse-field Ising model Hypr(g). The figure shows the energy (in an arbitrary unit) after projecting to the eigenstates vs. g for the 5-qubit (top panel (a)) and 6-qubit (bottom panel (b)) transverse-field Ising model. The procedure starts with two different initial states: (1) [(blue) dots that start on the lowest curve] the ground state |00000) of Hrrr(g = 0), and (2) [(red) dots that start on the top curve] the highest-energy state |11111) of Ayri(g = 0).
FIG. 12: Application of our spectral projection al- gorithm in the quantum annealing as the subroutine that carries out the measurement to project to eigen- states [3]. (a) The top figure shows the energy (in an arbitrary unit) after projecting to the eigenstates vs. g. Different colors represent different qubit numbers Ng = 14,16,18,20,&22. The procedure starts with the ground state |00---0) of Hrri(g = 0). (b) The bottom figures shows the energy variance (in an arbi- trary unit) at each step of projection (showing only for N, = 14,18, &22 for illustration), which can be used as a figure of merit for the error in the energy. Generally, the variance is the largest around g = 0.5, which is the critical point of the model in the thermodynamic imit. The curves are drawn to connect dots and to guide the eye. There are in total 210 steps in each projection run. The parameter At is chosen from the ist with number of repetitions shown in the parenthesis: 0.01(x 10), 0.1(« 10), 0.03( x50), 0.01( x 100), 0.003( x 40)] The ancillary state is chosen as (a = 1/2, B = e’?/V/2) with ¢ chosen randomly in [0, 27).
FIG. 12: Application of our spectral projection al- gorithm in the quantum annealing as the subroutine that carries out the measurement to project to eigen- states [3]. (a) The top figure shows the energy (in an arbitrary unit) after projecting to the eigenstates vs. g. Different colors represent different qubit numbers Ng = 14,16,18,20,&22. The procedure starts with the ground state |00---0) of Hrri(g = 0). (b) The bottom figures shows the energy variance (in an arbi- trary unit) at each step of projection (showing only for N, = 14,18, &22 for illustration), which can be used as a figure of merit for the error in the energy. Generally, the variance is the largest around g = 0.5, which is the critical point of the model in the thermodynamic imit. The curves are drawn to connect dots and to guide the eye. There are in total 210 steps in each projection run. The parameter At is chosen from the ist with number of repetitions shown in the parenthesis: 0.01(x 10), 0.1(« 10), 0.03( x50), 0.01( x 100), 0.003( x 40)] The ancillary state is chosen as (a = 1/2, B = e’?/V/2) with ¢ chosen randomly in [0, 27).
FIG. 13: The effect of decoherence on the quantum annealing. We use the contrived scenario that the deco- herence with ¢€ = 0.01 occurs at every 31th step in our spectral projection subroutine, where the primitive is run for 180 steps. We use the 5-qubit transverse-field Ising model Hyri(g) and carry out 3 different runs. The en- ergy (a) [top] and its variance (b) [bottom] are shown as g is varied. Both the energy and its variance are displayed in arbitary units. The initial state is the ground state |00000) of Hrri(g = 0). The final grounds are doubly degenerate and are | + — + —+) and | — +-—+-).
FIG. 13: The effect of decoherence on the quantum annealing. We use the contrived scenario that the deco- herence with ¢€ = 0.01 occurs at every 31th step in our spectral projection subroutine, where the primitive is run for 180 steps. We use the 5-qubit transverse-field Ising model Hyri(g) and carry out 3 different runs. The en- ergy (a) [top] and its variance (b) [bottom] are shown as g is varied. Both the energy and its variance are displayed in arbitary units. The initial state is the ground state |00000) of Hrri(g = 0). The final grounds are doubly degenerate and are | + — + —+) and | — +-—+-).
FIG. 15: The first one hundred values of r,,, starting with r; = 1. where it is convenient to define R, and a related Rp:
FIG. 15: The first one hundred values of r,,, starting with r; = 1. where it is convenient to define R, and a related Rp:
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