4.2 The exact weight-two benchmark
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4.2.1 Purpose and status
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The benchmark record compares a full state-vector simulator with a simulator restricted to the exact weight-two packet. Both use the same local gate arithmetic. The benchmark is not a hardware claim and not a universal performance model. It shows that the exact dimension reduction can translate into large memory and repeated-application gains in an ordinary implementation.
Recorded computation
At \(n=20\), the full state has \(1{,}048{,}576\) amplitudes and the exact weight-two packet has \(190\). The archived repeated-application gain is \(958.8\), the one-shot gain is \(54.8\), and the amplitude-memory ratio is \(5518.8\). The recorded relative reconstruction error is zero at the displayed precision.
4.2.2 Recorded table
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| \(n\) | full dim. | packet dim. | apply gain | one-shot gain | memory gain | rel. error |
|---|---|---|---|---|---|---|
| 8 | 256 | 28 | 1.11 | 1.64 | 9.14 | 0 |
| 10 | 1,024 | 45 | 1.51 | 2.02 | 22.8 | 0 |
| 12 | 4,096 | 66 | 2.95 | 3.98 | 62.1 | 0 |
| 14 | 16,384 | 91 | 8.55 | 11.0 | 180.0 | 0 |
| 16 | 65,536 | 120 | 33.0 | 36.9 | 546.1 | 0 |
| 18 | 262,144 | 153 | 131.3 | 39.1 | 1,713.4 | 0 |
| 20 | 1,048,576 | 190 | 958.8 | 54.8 | 5,518.8 | 0 |
4.2.3 What is machine independent
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The dimension identity
is independent of implementation. It predicts an exponential memory advantage for fixed \(k=2\). The measured runtime ratio need not equal the dimension ratio. Packet indexing, compilation, sparse update kernels, allocation, and cache behavior determine the constant factors.
4.2.4 Repeated application versus one shot
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Let \(C_{\rm compile}\) be the cost of constructing the packet basis and restricted gate plans, and let \(C_{\rm apply}\) be one packet application. Reusing the compiled plan \(R\) times costs
If the full path costs \(RC_{\rm full}\), the packet method breaks even when
This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
4.2.5 Independent reconstruction check
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The release code generates a random packet state, embeds it in the full Hilbert space, applies the same \(R_Z\) and exchange gates through both arithmetic paths, and compares the final states. The representative checks at \((n,k)=(6,1),(8,2),(10,2)\) produce errors near machine precision.
Scope.
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The recorded benchmark supports exactness and implementation value for a symmetry-preserving subclass. It does not show that generic quantum circuits admit comparable reductions.