Quantum Simulation of High Energy Physics (a non-Abelian lattice gauge theory)

[indrakshir]

Name

Simulating SU(2) Hadron dynamics on a 60 site lattice using 120 qubits

Circuit

SU(2)Hadron-LSH-x_100_meson

Observable value

59.99402699

Error bound (low)

-0.005710650099

Error bound (high)

+0.005710650099

Method

Loop-string-hadron encoding into qubits, 1st order Trotterization, measurement error mitigation, measurement in Z-basis. differential measurement protocol.

Method proof

Preprint of the work is available at:https://arxiv.org/pdf/2602.18080.

The reported instance involves a circuit of density (2-qubit gates/qubit pair) 0.91.

Authors

Fran Ilcic, Ritajit Majumdar, Emil Mathew, Md. Osama Ali, Nathan Earnest-Noble, Indrakshi Raychowdhury

Institutions

Birla Institute of Technology and Science Pilani, K K Birla Goa Campus,Goa, India; IBM Quantum, IBM India Research Lab, India, IBM Quantum, IBM T.J. Watson Research Center, USA.

Quantum runtime (seconds)

20

Classical runtime (seconds)

2485

Compute resources (quantum)

IBM Boston

Compute resources (classical)

NVIDIA RTX 5000 Ada GPU(AD102GL)

Notes

Only one instance is submitted here. The reported result is out come of both the circuits SCV and Meson, and the difference between observables are calculated to obtain physical results.

[Roland-Farrell]

Hi, we apologize for the delay in the review of this submission. We are currently adapting how submissions are categorized in the tracker and design changes will soon reflect this. Submissions will be categorized in three categories:

1. Baseline Benchmarks
   These represent problems of interest with sufficient documentation and scale and can include problems where the state-of-the-art is clearly classical, tracked for progress and calibration.
2. Active Candidates
   Problems that have met a higher bar of evidence suggesting quantum methods are credible contenders for advantage. To be clear, problems in this category are not provably examples of quantum advantage. These are merely problems where we do not yet have reliable classical benchmarks
3. Superseded Candidates
   Previously active candidates for which classical methods have caught up or exceeded the quantum approach.

Can you please suggest where you would like to place this submission? Your submission will be reviewed accordingly.

Furthermore, for making submissions more accessible, we are also requiring at least a 1 page write-up including the problem statement, a short abstract, summary of methods used and an illustration of the main result (for instance, graphical comparison of quantum and classical solutions/runtimes). This could be further supplemented by the additional details in your arXiv preprint.
Thanks

[indrakshir]

@Roland-Farrell, thanks for your comment. I would like to place the work at 2. Active Candidate. While this work does not demonstrate a clear advantage, it provides a direction toward practical quantum advantage for a class of problems in fundamental physics. I've attached a 2-page summary of the work (I'm sorry I couldn't fit it on one page). Looking forward to your comments.

LSH-IBM-advantage-tracker.pdf

[Roland-Farrell]

Hi @indrakshir ,

Thank you for providing the 2-page summary, it is quite helpful. The scale of your quantum simulations are certainly impressive, and simulations along this vein are very exciting directions for quantum advantage that is useful for high energy physics.

There is some additional information that would be helpful to add to your submission:

1. The Hamiltonian written in terms of Pauli operators acting on qubits, along with the couplings that are used in the quantum simulations corresponding to the submission. This can be found in your preprint but it would be nice to have it with the submission.
2. The initial states being prepared. If I understand correctly, you are evolving both the strong coupling vacuum (010101...) and a strong coupling meson (010111110101...). These correspond to computational basis states so it would be helpful to write them explicitly.
3. The observables being measured in terms of Pauli operators.
4. The value of the observable obtained from ibm_boston and a description for how the error bars are computed. Our current policy is that only "rigorous" error bars are reported. By rigorous, we mean that there should be some expectation that the noiseless value lies within the interval with e.g. 68% confidence. If I understand correctly, the error bars you are reporting are statistical (i.e. shot noise) and therefore would not meet this criteria.
5. A copy of the quantum circuit(s) used to compute the observable so that other people can test their classical and quantum methods. Additionally, please provide the raw data from which the observable is computed.

Thank you

[indrakshir]

Hi @Roland-Farrell , thanks for your encouraging comments. I'm attaching the addition as a separate README file, and have also created a pull request to include it in our submission.

I'm pasting the README file here as well.

Please let us know if you need any other details.

Best
Indrakshi.

README.txt

Quantum Advantage Tracker submission update
Instance: su2_lsh_hadron_dynamics_N60_x100_dt0p0015_t20

This README is an updated submission note prepared in response to the tracker
review comments. It supplements the earlier summary with:
(1) the qubit Hamiltonian used in the submitted quantum simulation,
(2) explicit computational-basis initial states,
(3) observables written in Pauli/Z language,
(4) the IBM Boston observable value for the submitted instance,
(5) the comparative error-bar prescription actually used in the work, and
(6) a reproducibility manifest for circuit files and raw data.

========================================================================
SHORT RESPONSE SUMMARY TO THE REVIEW POINTS

(1) Hamiltonian in qubit/Pauli form:
Added in Section 2.

(2) Explicit computational-basis initial states:
Added in Section 3.

(3) Observables in Pauli/Z language:
Added in Section 4.

(4) IBM Boston observable values and error-bar clarification for the submitted
instances t = 5, 10, 15, 20:
Added in Sections 5, 6, and 7. The README now explicitly states that the
quoted error bars are inter-method comparative estimates combined with an
exactly known symmetry-violation diagnostic, not shot-noise bars.

(5) Copies of circuits, raw data, and a circuit figure from the methods section:
Submission-package manifest added in Sections 8 and 11.

========================================================================
HOW ERROR BARS ARE TREATED IN THIS SUBMISSION

The quoted error bars in this submission are NOT statistical shot-noise error
bars.

For the dynamical observable n_f(t), the exact noiseless value is not known at
the scale of the 60-site / 120-qubit experiment. Therefore, within the present
study, the only principled uncertainty estimate available is a comparative
inter-method estimate, supplemented by an error diagnostic from violation of an
exactly conserved quantity.

Concretely:

• the observable n_f(t) is computed independently using four methods:
  TN, PP_CPU, PP_GPU, and QPU;
• the spread across these four values measures inter-method uncertainty;
• the conserved global charge Q is exactly known in the chosen sector;
• deviation of the measured/simulated Q from its exact value gives a direct
  diagnostic of error in that measurement;
• the final quoted error estimate combines the two in quadrature.

Thus, the uncertainty quoted here is a comparative cross-method error estimate
with symmetry-violation diagnosis built in. It is not a shot-noise bar.

========================================================================

1. PROBLEM INSTANCE
   ========================================================================

We study real-time dynamics in a (1+1)-dimensional SU(2) lattice gauge theory
with staggered fermions, encoded in the gauge-invariant Loop-String-Hadron
(LSH) basis and implemented on IBM Boston.

Submitted instance:

• system size: N = 60 lattice sites
• qubits: 120 (2 qubits per site)
• coupling: x = 100
• mass ratio: m/g = 1
• Trotter step size: dt = 0.0015
• number of Trotter steps in the original submitted instance: 20
  (this reply includes instances for time steps 5, 10, 15 in addition to 20)
• hardware backend: IBM Boston (156-qubit Heron processor)
• shot budget: 10,000 shots per Trotter step
• readout mitigation: measurement-error mitigation only

The main reported dynamical observable is built from a differential protocol:
one run evolves the strong-coupling vacuum (SCV), a second run evolves a
centered meson in the same background, and the reported signal is the
difference of the corresponding observables.

========================================================================
2. HAMILTONIAN AND QUBIT MAPPING

2.1 Dimensionless Hamiltonian

The scaled SU(2) Kogut-Susskind Hamiltonian is written as

W = H_E + mu H_M + x H_I

with
x = 1 / (g^2 a^2)
mu = 2 (m/g) sqrt(x)

For the submitted instance x = 100 and m/g = 1, so

mu = 20(sqrt(x))

The quantum circuit uses the large-x / weak-coupling LSH approximation discussed
in the paper and methods supplement.

2.2 LSH variables and qubit encoding

At each site r there are two binary LSH string-end occupations:
n_i(r) in {0,1}
n_o(r) in {0,1}

These are encoded directly in qubits. In operator form,

n_i(r) = (I - Z_i(r)) / 2
n_o(r) = (I - Z_o(r)) / 2

where Z_i(r) and Z_o(r) are Pauli-Z operators on the two qubits associated with
site r.

The loop variable n_l(r) is not stored as a separate qubit in this 1D setup; in
the weak-coupling approximation it is replaced by a large incoming boundary flux
l_i and treated analytically.

2.3 Approximate Hamiltonian used for the quantum circuit

The approximate electric term is

H_E^(approx) = const + A sum_r P_01(r),

where
A = l_i / 2 + 3/4

and P_01(r) is the projector onto the local two-qubit basis state
|n_i(r), n_o(r)> = |0,1>.

In Pauli form,

P_01(r) = [(I + Z_i(r)) (I - Z_o(r))] / 4
        = [I - Z_o(r) + Z_i(r) - Z_i(r) Z_o(r)] / 4.

So, up to an irrelevant global constant,

H_E^(approx) = (A/4) sum_r [I - Z_o(r) + Z_i(r) - Z_i(r) Z_o(r)].

The approximate mass term is

H_M^(approx) = sum_r (-1)^r [n_i(r) + n_o(r)].

Since n = (I - Z)/2 and N = 60 is even, this is equivalently

H_M^(approx) = -(1/2) sum_r (-1)^r [Z_i(r) + Z_o(r)].

The approximate interaction term is

H_I^(approx)
  = sum_r [ chi_o^+(r) chi_o^-(r+1) + chi_o^+(r+1) chi_o^-(r)
          + chi_i^+(r) chi_i^-(r+1) + chi_i^+(r+1) chi_i^-(r) ].

After Jordan-Wigner mapping this becomes an XY hopping Hamiltonian on the i and
o chains. In Pauli form,

H_I^(approx)
  = - sum_{r=0}^{N-2} [ X_i(r) X_i(r+1) + Y_i(r) Y_i(r+1) ]
    - sum_{r=0}^{N-2} [ X_o(r) X_o(r+1) + Y_o(r) Y_o(r+1) ].

Equivalently, in the logical ordering where the first N qubits are the i chain
and the next N qubits are the o chain,

H_I^(approx)
  = - sum_{r != N-1} [ X_r X_{r+1} + Y_r Y_{r+1} ].

This is the Pauli Hamiltonian implemented by the digital circuit up to the
large-x approximation and the omission of global phases.

2.4 Numerical couplings used in the submitted x = 100 simulation

For the first-order Trotter step,
dt = 0.0015

The interaction block uses
c = dt * x = 0.0015 * 100 = 0.15.

The mass block uses
m_tilde = dt * mu = 0.0015 * 20 = 0.03.

The electric block uses a site-pair phase
theta = -dt * (l_i/2 + 3/4),
with the exact numerical theta fixed by the chosen large-flux approximation and
encoded directly in the submitted circuit files.

The Trotter step is applied in the order

exp[-i m_tilde H_M] exp[-i dt H_E] exp[-i c H_I]

using first-order Lie-Trotter decomposition.

========================================================================
3. INITIAL STATES (COMPUTATIONAL BASIS STATES)

The submission uses two product states, both in the same global charge sector.

3.1 Strong-coupling vacuum (SCV)

In site language, the SCV is

• for even sites r: (n_i(r), n_o(r)) = (0,0)
• for odd sites r: (n_i(r), n_o(r)) = (1,1)

This gives n_f(r) = 0 on every site.

In per-site-pair notation
[(n_i(1),n_o(1)), (n_i(2),n_o(2)), ...]
the SCV is

(00,11,00,11,00,11,...,00,11).

In the initialization ordering used for the first Trotter step,
[n_i(1), n_i(2), n_o(1), n_o(2), n_i(3), n_i(4), n_o(3), n_o(4), ...],
the SCV is the computational-basis bitstring

01010101...0101

(i.e. repeating "0101").

In the simple zigzag ordering
[n_i(1), n_o(1), n_i(2), n_o(2), ...],
the same state is

00110011...0011

(i.e. repeating "0011").

The chosen initial ordering results in one fewer layer of SWAP gates in the time-evolution circuit.

3.2 Centered hadron (Baryon-anti-baryon) state

The hadron state is obtained from the SCV by placing hadronic excitations
on the two central staggered sites.

In site language (1-based site labels), the centered hadron is chosen as

• site 30 (even): (n_i, n_o) = (1,1)
• site 31 (odd): (n_i, n_o) = (0,0)

with all other sites identical to the SCV.

This gives
n_f(30) = 2,
n_f(31) = 2,
and n_f(r) = 0 elsewhere.

In per-site-pair notation this is

(00,11,...,00,11, 11,00, 00,11,...,00,11).

In the initialization ordering
[n_i(1), n_i(2), n_o(1), n_o(2), ...],
the state is again the SCV bitstring except for the central 4-bit block
corresponding to sites 30 and 31:

• SCV central block: 0101
• Hadron central block: 1100

One can also start with a central block 0110.
So the 120-bit initialization string is the same as the SCV except that the
central "0101" is replaced by "0110".

If one instead uses the charge-conjugate/reflected hadron orientation, the
central block would be "1001". These two choices are symmetry-related. The
exact choice used in the submission is fixed by the supplied circuit files.

All initial states are computational-basis product states and are prepared using
only X gates on the qubits whose bit value is 1.

========================================================================
4. OBSERVABLES IN PAULI FORM

4.1 Qubit occupancies

For each site r,

n_i(r) = (I - Z_i(r)) / 2
n_o(r) = (I - Z_o(r)) / 2

These are obtained experimentally from

P(1) = [1 - <Z>] / 2

for each qubit.

4.2 Local staggered fermion number

The local fermion number is staggered:

• on even sites:
  n_f(r) = n_i(r) + n_o(r)
  = 1 - [Z_i(r) + Z_o(r)] / 2

• on odd sites:
  n_f(r) = 2 - [n_i(r) + n_o(r)]
  = 1 + [Z_i(r) + Z_o(r)] / 2

Thus n_f(r) is a diagonal Pauli observable built entirely from local Z
expectation values.

4.3 Differential observable used in the experiment

The reported site-resolved signal is

Delta n_f(r,t) = n_f^hadron(r,t) - n_f^SCV(r,t).

This differential measurement protocol cancels part of the hardware bias and
removes the background dynamical contribution of the vacuum.

4.4 Scalar observable used in the present tracker instance

For the present tracker instance, we report the scalar trace

n_f(t) = sum_{r=1}^{60} Delta n_f(r,t),

evaluated at Trotter step t = 20.

4.5 Conserved global charge

The exactly conserved charge used as a diagnostic is

Q = sum_r [ n_i(r) + n_o(r) ] = 60

for the symmetry sector used here.

In Pauli form,

Q = 60 - (1/2) sum_r [ Z_i(r) + Z_o(r) ].

========================================================================
5. IBM BOSTON VALUES FOR THE SUBMITTED INSTANCES

The tracker-facing scalar observable is reported for four explicit circuit
instances at x = 100 and dt = 0.0015, corresponding to Trotter steps
t = 5, 10, 15, and 20. For each instance, the measured QPU value is listed
together with the TN, PP_CPU, and PP_GPU benchmark values and the
charge-violation diagnostics from the exactly known conserved quantity Q = 60.

========================================================================
5A. IBM BOSTON VALUE AND BENCHMARK DATA FOR t = 5

Submitted scalar observable:

• instance: x = 100, dt = 0.0015, Trotter step = 5
• backend: IBM Boston
• measured value:
  n_f(5) = -0.3550984939

Independent benchmark values for the same observable at step 5:

• TN = -0.4957689977
• PP_CPU = -0.4510787869
• PP_GPU = -0.47774275
• QPU = -0.3550984939

Because the exact value of n_f(5) is not known at this scale, the uncertainty
assigned to this observable is obtained by comparing the four independent method
estimates above, together with a diagnostic based on violation of the exactly
known conserved charge Q.

For the same step, the conserved-charge values are:

• TN = 60.00000000
• PP_CPU = 59.99991818
• PP_GPU = 60.00016050
• QPU = 59.95749581

The exact conserved value in this sector is

  Q = 60.

Hence the corresponding charge-violation diagnostics are:

• Delta Q_TN(5) = 0.00000000
• Delta Q_PP_CPU(5) = -0.00008182
• Delta Q_PP_GPU(5) = 0.00016050
• Delta Q_QPU(5) = -0.04250419

Equivalently, for the IBM Boston data,

  |Delta Q_QPU(5)| = |59.95749581 - 60| = 0.04250419.

This quantity is not itself the quoted error bar on n_f(5). Rather, it is the
symmetry-violation component used in the full comparative error estimate.

Comparative error estimate for t = 5:

• inter-method spread:
  sigma_std(5) = 0.062633290079
• charge-diagnostic aggregation:
  Delta Q_rms(5) = 0.021252285891
  Delta Q_rms(5)/sqrt(60) = 0.002743658311
• final shared comparative error estimate:
  sigma_shared(5) = 0.062693354409

========================================================================
5B. IBM BOSTON VALUE AND BENCHMARK DATA FOR t = 10

Submitted scalar observable:

• instance: x = 100, dt = 0.0015, Trotter step = 10
• backend: IBM Boston
• measured value:
  n_f(10) = 0.1431540203

Independent benchmark values for the same observable at step 10:

• TN = 0.3744026048
• PP_CPU = 0.4805343441
• PP_GPU = 0.394226
• QPU = 0.1431540203

Because the exact value of n_f(10) is not known at this scale, the uncertainty
assigned to this observable is obtained by comparing the four independent method
estimates above, together with a diagnostic based on violation of the exactly
known conserved charge Q.

For the same step, the conserved-charge values are:

• TN = 60.00000000
• PP_CPU = 59.99959358
• PP_GPU = 59.99729810
• QPU = 60.02942661

The exact conserved value in this sector is

  Q = 60.

Hence the corresponding charge-violation diagnostics are:

• Delta Q_TN(10) = 0.00000000
• Delta Q_PP_CPU(10) = -0.00040642
• Delta Q_PP_GPU(10) = -0.00270190
• Delta Q_QPU(10) = 0.02942661

Equivalently, for the IBM Boston data,

  |Delta Q_QPU(10)| = |60.02942661 - 60| = 0.02942661.

This quantity is not itself the quoted error bar on n_f(10). Rather, it is the
symmetry-violation component used in the full comparative error estimate.

Comparative error estimate for t = 10:

• inter-method spread:
  sigma_std(10) = 0.144177140195
• charge-diagnostic aggregation:
  Delta Q_rms(10) = 0.014776593120
  Delta Q_rms(10)/sqrt(60) = 0.001907649969
• final shared comparative error estimate:
  sigma_shared(10) = 0.144189759980

========================================================================
5C. IBM BOSTON VALUE AND BENCHMARK DATA FOR t = 15

Submitted scalar observable:

• instance: x = 100, dt = 0.0015, Trotter step = 15
• backend: IBM Boston
• measured value:
  n_f(15) = 0.00494274374

Independent benchmark values for the same observable at step 15:

• TN = -0.1131674269
• PP_CPU = -0.03107728475
• PP_GPU = -0.096355725
• QPU = 0.00494274374

Because the exact value of n_f(15) is not known at this scale, the uncertainty
assigned to this observable is obtained by comparing the four independent method
estimates above, together with a diagnostic based on violation of the exactly
known conserved charge Q.

For the same step, the conserved-charge values are:

• TN = 60.00000000
• PP_CPU = 60.01033665
• PP_GPU = 60.00620690
• QPU = 59.91796245

The exact conserved value in this sector is

  Q = 60.

Hence the corresponding charge-violation diagnostics are:

• Delta Q_TN(15) = 0.00000000
• Delta Q_PP_CPU(15) = 0.01033665
• Delta Q_PP_GPU(15) = 0.00620690
• Delta Q_QPU(15) = -0.08203755

Equivalently, for the IBM Boston data,

  |Delta Q_QPU(15)| = |59.91796245 - 60| = 0.08203755.

This quantity is not itself the quoted error bar on n_f(15). Rather, it is the
symmetry-violation component used in the full comparative error estimate.

Comparative error estimate for t = 15:

• inter-method spread:
  sigma_std(15) = 0.055371122250
• charge-diagnostic aggregation:
  Delta Q_rms(15) = 0.041459412535
  Delta Q_rms(15)/sqrt(60) = 0.005352387143
• final shared comparative error estimate:
  sigma_shared(15) = 0.055629211996

========================================================================
5D. IBM BOSTON VALUE AND BENCHMARK DATA FOR t = 20

Submitted scalar observable:

• instance: x = 100, dt = 0.0015, Trotter step = 20
• backend: IBM Boston
• measured value:
  n_f(20) = -0.07723426024

Independent benchmark values for the same observable at step 20:

• TN = 0.2778348725
• PP_CPU = 0.4081775866
• PP_GPU = 0.2456276125
• QPU = -0.07723426024

Because the exact value of n_f(20) is not known at this scale, the uncertainty
assigned to this observable is obtained by comparing the four independent method
estimates above, together with a diagnostic based on violation of the exactly
known conserved charge Q.

For the same step, the conserved-charge values are:

• TN = 60.00000000
• PP_CPU = 60.00538161
• PP_GPU = 59.99313828
• QPU = 59.99402699

The exact conserved value in this sector is

  Q = 60.

Hence the corresponding charge-violation diagnostics are:

• Delta Q_TN(20) = 0.00000000
• Delta Q_PP_CPU(20) = 0.00538161
• Delta Q_PP_GPU(20) = -0.00686172
• Delta Q_QPU(20) = -0.00597301

Equivalently, for the IBM Boston data,

  |Delta Q_QPU(20)| = |59.99402699 - 60| = 0.00597301.

This quantity is not itself the quoted error bar on n_f(20). Rather, it is the
symmetry-violation component used in the full comparative error estimate.

Comparative error estimate for t = 20:

• inter-method spread:
  sigma_std(20) = 0.206233598084
• charge-diagnostic aggregation:
  Delta Q_rms(20) = 0.005284926111
  Delta Q_rms(20)/sqrt(60) = 0.000682281027
• final shared comparative error estimate:
  sigma_shared(20) = 0.206234726673

========================================================================
6. ERROR-BAR PRESCRIPTION USED IN THIS WORK

The error bars quoted for the dynamical observable n_f(t) are defined by a
comparative inter-method prescription together with an exactly known symmetry
diagnostic. This is the prescription used in the methods supplement.

6.1 Four independent estimates of the observable

For each time slice t, we obtain four estimates

n_f^(m)(t),   m in {TN, PP_CPU, PP_GPU, QPU}.

Because the exact value of n_f(t) is not known at this scale, these four values
are treated as independent method estimates of the same quantity.

6.2 Inter-method spread

The methods supplement defines the inter-method spread using the mean across the
four estimates:

mean_n_f(t) = (1/4) sum_m n_f^(m)(t)

sigma_std(t) = sqrt[ (1/3) sum_m ( n_f^(m)(t) - mean_n_f(t) )^2 ].

This term quantifies the method-to-method spread of the observable.

6.3 Symmetry-violation diagnostic from the exactly known conserved charge

The global charge is exactly known:

Q = sum_r [n_i(r) + n_o(r)] = 60.

For each method m, define the charge violation

Delta Q^(m)(t) = Q^(m)(t) - 60.

Since Q is a sum over L = 60 sites, the methods supplement converts this into a
conservative per-site diagnostic scale and then aggregates over methods through

Delta Q_rms(t) = sqrt[ (1/4) sum_m (Delta Q^(m)(t))^2 ].

The corresponding per-site symmetry-violation scale is

Delta Q_rms(t) / sqrt(60).

This term is directly grounded in an exactly known physical constraint.

6.4 Final shared error estimate

The final shared uncertainty band used in the methods supplement is the
quadrature combination

sigma_shared(t)
  = sqrt[ sigma_std(t)^2 + ( Delta Q_rms(t) / sqrt(60) )^2 ].

This is the comparative error estimate quoted for the observable. It combines:

• inter-method spread in the observable itself, and
• diagnosed symmetry violation through the exactly known conserved charge.

6.5 Method-specific error estimate

The supplement also defines an individual method-specific error bar

sigma_m(t)
  = sqrt[ sigma_pair,m(t)^2 + ( |Delta Q^(m)(t)| / sqrt(60) )^2 ],

where

sigma_pair,m(t) = | n_f^(m)(t) - n_ref(t) |.

This is the per-method comparative error estimate.

6.6 Important clarification

These quoted error bars are NOT statistical shot-noise bars. They are not
derived from repeated sampling around a known exact observable value. Instead,
they are the uncertainty estimates used in this work precisely because:

• the exact value of n_f(t) is not known at the present scale,
• multiple classical/quantum methods are available for comparison, and
• the exactly known conserved quantity Q provides a direct diagnostic of
  measurement/simulation error.

========================================================================
7. NUMERICAL ERROR-ESTIMATE SUMMARY FOR THE SUBMITTED INSTANCES

The table below collects the comparative error-estimate ingredients for the four
explicitly reported circuit instances.

Step sigma_std(t) DeltaQ_rms/sqrt(60) sigma_shared(t) |DeltaQ_QPU|
5 0.062633290079 0.002743658311 0.062693354409 0.04250419
10 0.144177140195 0.001907649969 0.144189759980 0.02942661
15 0.055371122250 0.005352387143 0.055629211996 0.08203755
20 0.206233598084 0.000682281027 0.206234726673 0.00597301

Thus, for all four submitted instances, the quoted uncertainty is dominated by
the inter-method spread among TN / PP_CPU / PP_GPU / QPU values, while the
symmetry-violation term from the exactly known conserved charge provides a
direct additional diagnostic.

========================================================================
8. QUANTUM CIRCUITS USED

The submission requires two circuit families for each reported Trotter step:
(1) SCV evolution circuits
(2) centered-hadron evolution circuits

Because the observable is differential, both circuits are required to reproduce
the reported data.

The explicit submitted circuit instances are:

• t = 5
• t = 10
• t = 15
• t = 20

Circuit structure:

• first-order Trotterization
• interaction block for H_I^(approx)
• electric block for H_E^(approx)
• mass block for H_M^(approx)
• SWAP layers to mediate nearest-neighbor interactions in the hardware layout
• measurement of all 120 qubits in the Z basis

Depth information:

• two-qubit depth for t Trotter steps: 13 t - 1
• for t = 5: two-qubit depth = 64
• for t = 10: two-qubit depth = 129
• for t = 15: two-qubit depth = 194
• for t = 20: two-qubit depth = 259
• for reference, at t = 25 the circuit has two-qubit depth 324,
  17,660 two-qubit gates, and 90,955 total gates

A cropped version of the circuit figure from the methods section is included
with this submission package as:

• figures/circuit_figure_methods.png

========================================================================
9. CLASSICAL BENCHMARK METHODS

9.1 Tensor Network (TN)

The TN benchmark evolves the full LSH Hamiltonian using an MPS/MPO
representation and 2-site TDVP, with maximum bond dimension 200 and bosonic
cutoff j_max = 5/2.

For x = 100, the maximum reported TN truncation error in the Schmidt spectrum
reaches 1e-4 by Trotter step 20.

9.2 Pauli Propagation on CPU (PP_CPU)

The CPU PP benchmark simulates the noiseless circuit in the Heisenberg picture.
For x = 100, the safe upper bound used for the non-truncated PP term count is
12,500 terms. This is treated as effectively exact up to floating-point
tolerances.

9.3 Pauli Propagation on GPU (PP_GPU)

The GPU PP benchmark uses NVIDIA cuQuantum / cuPauliProp with
coefficient-based truncation. The coefficient cutoff is

epsilon = 1e-5.

This is a controlled circuit-level approximation that substantially reduces
memory and runtime.

========================================================================
10. RUNTIME DATA FOR THE SUBMITTED INSTANCES

The supplementary runtime table for x = 100 gives the following values for the
explicitly reported instances:

Step QPU (s) TN (s) PP_CPU (s) PP_GPU (s)
5 20 584.092 477.4471 547.5810
10 20 4936.112 1111.2079 1186.1290
15 20 11300.992 1798.2811 1990.1430
20 20 8949.110 2484.3693 2930.4400

========================================================================
11. REPRODUCIBILITY FILE MANIFEST

The tracker review requested:

• a copy of the quantum circuits used to compute the observable
• the raw data from which the observable is computed

Accordingly, the final submission package should include the following files
(or files with equivalent content and different names):

11.1 Circuit files: available at https://github.com/indrakshir/LSH-IBM

SCV circuits:

• circuits/scv_N60_x100_dt0p0015_steps05.qasm
• circuits/scv_N60_x100_dt0p0015_steps10.qasm
• circuits/scv_N60_x100_dt0p0015_steps15.qasm
• circuits/scv_N60_x100_dt0p0015_steps20.qasm

Centered-hadron circuits:

• circuits/hadron_N60_x100_dt0p0015_steps05.qasm
• circuits/hadron_N60_x100_dt0p0015_steps10.qasm
• circuits/hadron_N60_x100_dt0p0015_steps15.qasm
• circuits/hadron_N60_x100_dt0p0015_steps20.qasm

11.2 Raw experimental data

Raw experimental data is available at: https://github.com/mathew0036/lsh_data

11.5 README note on exact observables

For the current submission observable n_f(r,t), per-qubit Z expectations are
sufficient because the observable is diagonal and linear in the single-site
occupancies. Full bitstring counts are nevertheless preferable because they
allow independent recomputation of additional observables.

========================================================================
12. SUBMITTED BENCHMARK TABLES

12.1 Scalar observable n_f(t)

Step TN PP_CPU PP_GPU QPU Median St.dev
1 1.824222901 1.825335615 1.825336 1.77264257 1.824779258 0.02616639397
2 1.343818565 1.34880517 1.347871 1.234719061 1.345844782 0.05609802779
3 0.6852977079 0.698419082 0.693895 0.7005505103 0.696157041 0.006757945893
4 0.01781852652 0.04434743353 0.0317106 0.1125808617 0.03802901676 0.04206360493
5 -0.4957689977 -0.4510787869 -0.47774275 -0.3550984939 -0.4644107685 0.0626332901
6 -0.7447406887 -0.6794112084 -0.725069325 -0.4792900767 -0.7022402667 0.1216756278
7 -0.6994223897 -0.6144203611 -0.68012595 -0.405844541 -0.6472731556 0.1344237824
8 -0.4149706965 -0.3149685829 -0.3959514 -0.3160420563 -0.3559967282 0.05251500138
9 -0.009710517986 0.09777330494 0.010149475 -0.01701975315 0.0002194785072 0.05291043151
10 0.3744026048 0.4805343441 0.394226 0.1431540203 0.3843143024 0.1441771402
11 0.6163184396 0.7136997525 0.6325666 0.3540697043 0.6244425198 0.1559910262
12 0.6514199805 0.7369940536 0.6619868 0.4092782333 0.6567033902 0.142288607
13 0.4873750145 0.5641770138 0.4951614125 0.2630479833 0.4912682135 0.1308944493
14 0.1962856389 0.2712616284 0.2069994 0.1957298805 0.2016425194 0.0361687867
15 -0.1131674269 -0.03107728475 -0.096355725 0.00494274374 -0.06371650487 0.05537112224
16 -0.3336240614 -0.2318235614 -0.314319025 -0.08874021367 -0.2730712932 0.1113808299
17 -0.3941897866 -0.2691116404 -0.3787042075 -0.1962503187 -0.323907924 0.09384077972
18 -0.2807460955 -0.1325140013 -0.272586375 -0.1673053457 -0.2199458603 0.07462290823
19 -0.03330018147 0.1236920782 -0.035842523 -0.1490836294 -0.03457135223 0.1120789393
20 0.2778348725 0.4081775866 0.2456276125 -0.07723426024 0.2617312425 0.2062335981

12.2 Conserved charge Q

Step TN PP_CPU PP_GPU QPU error_diagnostic
1 60 60 60 60.07076187 0.03538093432
2 60 60 60 60.01556783 0.007783914764
3 60 59.99999892 60.0000295 59.97441541 0.01279704071
4 60 59.99998537 60.000264 59.98574097 0.007172217667
5 60 59.99991818 60.0001605 59.95749581 0.02126544441
6 60 59.99985924 59.99994432 59.9228317 0.03855145367
7 60 59.99978099 60.0001445 59.97723002 0.0113735536
8 60 60.00033369 59.99693215 60.01516427 0.008182322904
9 60 60.00063853 59.99640159 60.09153313 0.04629746559
10 60 59.99959358 59.9972981 60.02942661 0.01527773734
11 60 60.00017695 60.0023345 59.95262885 0.02412750108
12 60 59.99506226 60.0064158 59.97553288 0.01331739931
13 60 59.99594391 60.00947415 59.95853503 0.02236597107
14 60 60.00326018 60.0093484 60.09813368 0.04712493522
15 60 60.01033665 60.0062069 59.91796245 0.04398168579
16 60 60.01345623 59.99751846 59.92227266 0.04129079
17 60 60.00953873 59.98460915 59.92904491 0.03599839525
18 60 59.99695801 59.9810714 60.07128097 0.04016899441
19 60 59.99513071 59.98358415 60.08671777 0.04740887609
20 60 60.00538161 59.99313828 59.99402699 0.005710650099

========================================================================

========================================================================
END OF README

I've attached a picture of the circuits us

ed. Corresponding QASM files are available publicly at: https://github.com/indrakshir/LSH-IBM

README_QAT_su2_lsh_hadron_dynamics_reply.txt