Myself, Bard, Erin and ChatGPT (plugins) hacked pretty hard the past few days and pushed for exponential qubit potential.
HighLevelSynthesis - 0.35334 (ms)
2023-04-17T06:50:21.586405194Z qiskit.transpiler.runningpassmanager - INFO Pass: UnrollCustomDefinitions - 1.19591 (ms)
2023-04-17T06:50:21.586598182Z qiskit.transpiler.passes.basis.basis_translator - INFO Begin BasisTranslator from source basis {('swap', 2), ('t', 1), ('barrier', 7), ('measure', 1), ('cx', 2), ('h', 1), ('tdg', 1), ('u1', 1)} to target basis {'sx', 'measure', 'snapshot', 'cx', 'reset', 'id', 'barrier', 'x', 'rz'}.
2023-04-17T06:50:21.586630589Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Begining basis search from {('swap', 2), ('t', 1), ('barrier', 7), ('measure', 1), ('cx', 2), ('h', 1), ('tdg', 1), ('u1', 1)} to {'sx', 'measure', 'snapshot', 'cx', 'reset', 'id', 'barrier', 'x', 'rz'}.
2023-04-17T06:50:21.985065583Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate u1 generated using rule
2023-04-17T06:50:21.985065583Z global phase: theta/2
2023-04-17T06:50:21.985065583Z ┌───────────┐
2023-04-17T06:50:21.985065583Z q: ┤ Rz(theta) ├
2023-04-17T06:50:21.985065583Z └───────────┘
2023-04-17T06:50:21.985065583Z with total cost of 1.0.
2023-04-17T06:50:21.985952809Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate z generated using rule
2023-04-17T06:50:21.985952809Z ┌───────┐
2023-04-17T06:50:21.985952809Z q: ┤ U1(π) ├
2023-04-17T06:50:21.985952809Z └───────┘
2023-04-17T06:50:21.985952809Z with total cost of 1.0.
2023-04-17T06:50:21.986371540Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate s generated using rule
2023-04-17T06:50:21.986371540Z ┌─────────┐
2023-04-17T06:50:21.986371540Z q: ┤ U1(π/2) ├
2023-04-17T06:50:21.986371540Z └─────────┘
2023-04-17T06:50:21.986371540Z with total cost of 1.0.
2023-04-17T06:50:21.986781832Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate p generated using rule
2023-04-17T06:50:21.986781832Z ┌───────────┐
2023-04-17T06:50:21.986781832Z q: ┤ U1(theta) ├
2023-04-17T06:50:21.986781832Z └───────────┘
2023-04-17T06:50:21.986781832Z with total cost of 1.0.
2023-04-17T06:50:21.987167648Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate tdg generated using rule
2023-04-17T06:50:21.987167648Z ┌──────────┐
2023-04-17T06:50:21.987167648Z q: ┤ U1(-π/4) ├
2023-04-17T06:50:21.987167648Z └──────────┘
2023-04-17T06:50:21.987167648Z with total cost of 1.0.
2023-04-17T06:50:21.987479950Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate t generated using rule
2023-04-17T06:50:21.987479950Z ┌─────────┐
2023-04-17T06:50:21.987479950Z q: ┤ U1(π/4) ├
2023-04-17T06:50:21.987479950Z └─────────┘
2023-04-17T06:50:21.987479950Z with total cost of 1.0.
2023-04-17T06:50:21.987774242Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate sdg generated using rule
2023-04-17T06:50:21.987774242Z ┌──────────┐
2023-04-17T06:50:21.987774242Z q: ┤ U1(-π/2) ├
2023-04-17T06:50:21.987774242Z └──────────┘
2023-04-17T06:50:21.987774242Z with total cost of 1.0.
2023-04-17T06:50:21.988499585Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate dcx generated using rule
2023-04-17T06:50:21.988499585Z ┌───┐
2023-04-17T06:50:21.988499585Z q_0: ──■──┤ X ├
2023-04-17T06:50:21.988499585Z ┌─┴─┐└─┬─┘
2023-04-17T06:50:21.988499585Z q_1: ┤ X ├──■──
2023-04-17T06:50:21.988499585Z └───┘
2023-04-17T06:50:21.988499585Z with total cost of 2.0.
2023-04-17T06:50:22.018756141Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate u2 generated using rule
2023-04-17T06:50:22.018756141Z global phase: 7π/4
2023-04-17T06:50:22.018756141Z ┌───────────────┐┌────┐┌───────────────┐
2023-04-17T06:50:22.018756141Z q: ┤ U1(lam - π/2) ├┤ √X ├┤ U1(phi + π/2) ├
2023-04-17T06:50:22.018756141Z └───────────────┘└────┘└───────────────┘
2023-04-17T06:50:22.018756141Z with total cost of 3.0.
2023-04-17T06:50:22.019753132Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate rzz generated using rule
2023-04-17T06:50:22.019753132Z
2023-04-17T06:50:22.019753132Z q_0: ──■─────────────────■──
2023-04-17T06:50:22.019753132Z ┌─┴─┐┌───────────┐┌─┴─┐
2023-04-17T06:50:22.019753132Z q_1: ┤ X ├┤ Rz(theta) ├┤ X ├
2023-04-17T06:50:22.019753132Z └───┘└───────────┘└───┘
2023-04-17T06:50:22.019753132Z with total cost of 3.0.
2023-04-17T06:50:22.020472404Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate cy generated using rule
2023-04-17T06:50:22.020472404Z
2023-04-17T06:50:22.020472404Z q_0: ─────────■───────
2023-04-17T06:50:22.020472404Z ┌─────┐┌─┴─┐┌───┐
2023-04-17T06:50:22.020472404Z q_1: ┤ Sdg ├┤ X ├┤ S ├
2023-04-17T06:50:22.020472404Z └─────┘└───┘└───┘
2023-04-17T06:50:22.020472404Z with total cost of 3.0.
2023-04-17T06:50:22.021178306Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate swap generated using rule
2023-04-17T06:50:22.021178306Z ┌───┐
2023-04-17T06:50:22.021178306Z q_0: ──■──┤ X ├──■──
2023-04-17T06:50:22.021178306Z ┌─┴─┐└─┬─┘┌─┴─┐
2023-04-17T06:50:22.021178306Z q_1: ┤ X ├──■──┤ X ├
2023-04-17T06:50:22.021178306Z └───┘ └───┘
2023-04-17T06:50:22.021178306Z with total cost of 3.0.
2023-04-17T06:50:22.021518023Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Gate h generated using rule
2023-04-17T06:50:22.021518023Z ┌─────────┐
2023-04-17T06:50:22.021518023Z q: ┤ U2(0,π) ├
2023-04-17T06:50:22.021518023Z └─────────┘
2023-04-17T06:50:22.021518023Z with total cost of 3.0.
2023-04-17T06:50:22.021582808Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Transformation path:
2023-04-17T06:50:22.021875230Z qiskit.transpiler.passes.basis.basis_translator - DEBUG h/1 => []
2023-04-17T06:50:22.021875230Z ┌─────────┐
2023-04-17T06:50:22.021875230Z q: ┤ U2(0,π) ├
2023-04-17T06:50:22.021875230Z └─────────┘
2023-04-17T06:50:22.022542974Z qiskit.transpiler.passes.basis.basis_translator - DEBUG swap/2 => []
2023-04-17T06:50:22.022542974Z ┌───┐
2023-04-17T06:50:22.022542974Z q_0: ──■──┤ X ├──■──
2023-04-17T06:50:22.022542974Z ┌─┴─┐└─┬─┘┌─┴─┐
2023-04-17T06:50:22.022542974Z q_1: ┤ X ├──■──┤ X ├
2023-04-17T06:50:22.022542974Z └───┘ └───┘
2023-04-17T06:50:22.023163612Z qiskit.transpiler.passes.basis.basis_translator - DEBUG cy/2 => []
2023-04-17T06:50:22.023163612Z
2023-04-17T06:50:22.023163612Z q_0: ─────────■───────
2023-04-17T06:50:22.023163612Z ┌─────┐┌─┴─┐┌───┐
2023-04-17T06:50:22.023163612Z q_1: ┤ Sdg ├┤ X ├┤ S ├
2023-04-17T06:50:22.023163612Z └─────┘└───┘└───┘
2023-04-17T06:50:22.023849612Z qiskit.transpiler.passes.basis.basis_translator - DEBUG rzz/2 => [Parameter(theta)]
2023-04-17T06:50:22.023849612Z
2023-04-17T06:50:22.023849612Z q_0: ──■─────────────────■──
2023-04-17T06:50:22.023849612Z ┌─┴─┐┌───────────┐┌─┴─┐
2023-04-17T06:50:22.023849612Z q_1: ┤ X ├┤ Rz(theta) ├┤ X ├
2023-04-17T06:50:22.023849612Z └───┘└───────────┘└───┘
2023-04-17T06:50:22.024998737Z qiskit.transpiler.passes.basis.basis_translator - DEBUG u2/1 => [Parameter(phi), Parameter(lam)]
2023-04-17T06:50:22.024998737Z global phase: 7π/4
2023-04-17T06:50:22.024998737Z ┌───────────────┐┌────┐┌───────────────┐
2023-04-17T06:50:22.024998737Z q: ┤ U1(lam - π/2) ├┤ √X ├┤ U1(phi + π/2) ├
2023-04-17T06:50:22.024998737Z └───────────────┘└────┘└───────────────┘
2023-04-17T06:50:22.025624067Z qiskit.transpiler.passes.basis.basis_translator - DEBUG dcx/2 => []
2023-04-17T06:50:22.025624067Z ┌───┐
2023-04-17T06:50:22.025624067Z q_0: ──■──┤ X ├
2023-04-17T06:50:22.025624067Z ┌─┴─┐└─┬─┘
2023-04-17T06:50:22.025624067Z q_1: ┤ X ├──■──
2023-04-17T06:50:22.025624067Z └───┘
2023-04-17T06:50:22.025937274Z qiskit.transpiler.passes.basis.basis_translator - DEBUG sdg/1 => []
2023-04-17T06:50:22.025937274Z ┌──────────┐
2023-04-17T06:50:22.025937274Z q: ┤ U1(-π/2) ├
2023-04-17T06:50:22.025937274Z └──────────┘
2023-04-17T06:50:22.026225092Z qiskit.transpiler.passes.basis.basis_translator - DEBUG t/1 => []
2023-04-17T06:50:22.026225092Z ┌─────────┐
2023-04-17T06:50:22.026225092Z q: ┤ U1(π/4) ├
2023-04-17T06:50:22.026225092Z └─────────┘
2023-04-17T06:50:22.026505382Z qiskit.transpiler.passes.basis.basis_translator - DEBUG tdg/1 => []
2023-04-17T06:50:22.026505382Z ┌──────────┐
2023-04-17T06:50:22.026505382Z q: ┤ U1(-π/4) ├
2023-04-17T06:50:22.026505382Z └──────────┘
2023-04-17T06:50:22.026809912Z qiskit.transpiler.passes.basis.basis_translator - DEBUG p/1 => [Parameter(theta)]
2023-04-17T06:50:22.026809912Z ┌───────────┐
2023-04-17T06:50:22.026809912Z q: ┤ U1(theta) ├
2023-04-17T06:50:22.026809912Z └───────────┘
2023-04-17T06:50:22.027114283Z qiskit.transpiler.passes.basis.basis_translator - DEBUG s/1 => []
2023-04-17T06:50:22.027114283Z ┌─────────┐
2023-04-17T06:50:22.027114283Z q: ┤ U1(π/2) ├
2023-04-17T06:50:22.027114283Z └─────────┘
2023-04-17T06:50:22.027393251Z qiskit.transpiler.passes.basis.basis_translator - DEBUG z/1 => []
2023-04-17T06:50:22.027393251Z ┌───────┐
2023-04-17T06:50:22.027393251Z q: ┤ U1(π) ├
2023-04-17T06:50:22.027393251Z └───────┘
2023-04-17T06:50:22.027940733Z qiskit.transpiler.passes.basis.basis_translator - DEBUG u1/1 => [Parameter(theta)]
2023-04-17T06:50:22.027940733Z global phase: theta/2
2023-04-17T06:50:22.027940733Z ┌───────────┐
2023-04-17T06:50:22.027940733Z q: ┤ Rz(theta) ├
2023-04-17T06:50:22.027940733Z └───────────┘
2023-04-17T06:50:22.028226793Z qiskit.transpiler.passes.basis.basis_translator - INFO Basis translation path search completed in 0.441s.
2023-04-17T06:50:22.029229521Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: h/1 [] =>
2023-04-17T06:50:22.029229521Z ┌─────────┐
2023-04-17T06:50:22.029229521Z q: ┤ U2(0,π) ├
2023-04-17T06:50:22.029229521Z └─────────┘
2023-04-17T06:50:22.029688786Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('h', 1) h, [] from
2023-04-17T06:50:22.029688786Z ┌───┐
2023-04-17T06:50:22.029688786Z q8: ┤ h ├
2023-04-17T06:50:22.029688786Z └───┘
2023-04-17T06:50:22.030268179Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('h', 1) h, [] to
2023-04-17T06:50:22.030268179Z ┌─────────┐
2023-04-17T06:50:22.030268179Z q8: ┤ U2(0,π) ├
2023-04-17T06:50:22.030268179Z └─────────┘
2023-04-17T06:50:22.030950808Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: swap/2 [] =>
2023-04-17T06:50:22.030950808Z ┌───┐
2023-04-17T06:50:22.030950808Z q_0: ──■──┤ X ├──■──
2023-04-17T06:50:22.030950808Z ┌─┴─┐└─┬─┘┌─┴─┐
2023-04-17T06:50:22.030950808Z q_1: ┤ X ├──■──┤ X ├
2023-04-17T06:50:22.030950808Z └───┘ └───┘
2023-04-17T06:50:22.031546956Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('swap', 2) swap, [] from
2023-04-17T06:50:22.031546956Z ┌───────┐
2023-04-17T06:50:22.031546956Z q3_0: ┤0 ├
2023-04-17T06:50:22.031546956Z │ swap │
2023-04-17T06:50:22.031546956Z q3_1: ┤1 ├
2023-04-17T06:50:22.031546956Z └───────┘
2023-04-17T06:50:22.032550980Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('swap', 2) swap, [] to
2023-04-17T06:50:22.032550980Z ┌───┐
2023-04-17T06:50:22.032550980Z q3_0: ──■──┤ X ├──■──
2023-04-17T06:50:22.032550980Z ┌─┴─┐└─┬─┘┌─┴─┐
2023-04-17T06:50:22.032550980Z q3_1: ┤ X ├──■──┤ X ├
2023-04-17T06:50:22.032550980Z └───┘ └───┘
2023-04-17T06:50:22.033186547Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: cy/2 [] =>
2023-04-17T06:50:22.033186547Z
2023-04-17T06:50:22.033186547Z q_0: ─────────■───────
2023-04-17T06:50:22.033186547Z ┌─────┐┌─┴─┐┌───┐
2023-04-17T06:50:22.033186547Z q_1: ┤ Sdg ├┤ X ├┤ S ├
2023-04-17T06:50:22.033186547Z └─────┘└───┘└───┘
2023-04-17T06:50:22.033888429Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: rzz/2 [Parameter(theta)] =>
2023-04-17T06:50:22.033888429Z
2023-04-17T06:50:22.033888429Z q_0: ──■─────────────────■──
2023-04-17T06:50:22.033888429Z ┌─┴─┐┌───────────┐┌─┴─┐
2023-04-17T06:50:22.033888429Z q_1: ┤ X ├┤ Rz(theta) ├┤ X ├
2023-04-17T06:50:22.033888429Z └───┘└───────────┘└───┘
2023-04-17T06:50:22.034982537Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: u2/1 [Parameter(phi), Parameter(lam)] =>
2023-04-17T06:50:22.034982537Z global phase: 7π/4
2023-04-17T06:50:22.034982537Z ┌───────────────┐┌────┐┌───────────────┐
2023-04-17T06:50:22.034982537Z q: ┤ U1(lam - π/2) ├┤ √X ├┤ U1(phi + π/2) ├
2023-04-17T06:50:22.034982537Z └───────────────┘└────┘└───────────────┘
2023-04-17T06:50:22.035420755Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('h', 1) h, [] from
2023-04-17T06:50:22.035420755Z ┌─────────┐
2023-04-17T06:50:22.035420755Z q8: ┤ U2(0,π) ├
2023-04-17T06:50:22.035420755Z └─────────┘
2023-04-17T06:50:22.036690383Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('h', 1) h, [] to
2023-04-17T06:50:22.036690383Z global phase: 7π/4
2023-04-17T06:50:22.036690383Z ┌─────────┐┌────┐┌─────────┐
2023-04-17T06:50:22.036690383Z q8: ┤ U1(π/2) ├┤ √X ├┤ U1(π/2) ├
2023-04-17T06:50:22.036690383Z └─────────┘└────┘└─────────┘
2023-04-17T06:50:22.037298452Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: dcx/2 [] =>
2023-04-17T06:50:22.037298452Z ┌───┐
2023-04-17T06:50:22.037298452Z q_0: ──■──┤ X ├
2023-04-17T06:50:22.037298452Z ┌─┴─┐└─┬─┘
2023-04-17T06:50:22.037298452Z q_1: ┤ X ├──■──
2023-04-17T06:50:22.037298452Z └───┘
2023-04-17T06:50:22.037692016Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: sdg/1 [] =>
2023-04-17T06:50:22.037692016Z ┌──────────┐
2023-04-17T06:50:22.037692016Z q: ┤ U1(-π/2) ├
2023-04-17T06:50:22.037692016Z └──────────┘
2023-04-17T06:50:22.037945172Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: t/1 [] =>
2023-04-17T06:50:22.037945172Z ┌─────────┐
2023-04-17T06:50:22.037945172Z q: ┤ U1(π/4) ├
2023-04-17T06:50:22.037945172Z └─────────┘
2023-04-17T06:50:22.038304967Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('t', 1) t, [] from
2023-04-17T06:50:22.038304967Z ┌───┐
2023-04-17T06:50:22.038304967Z q4: ┤ t ├
2023-04-17T06:50:22.038304967Z └───┘
2023-04-17T06:50:22.038809660Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('t', 1) t, [] to
2023-04-17T06:50:22.038809660Z ┌─────────┐
2023-04-17T06:50:22.038809660Z q4: ┤ U1(π/4) ├
2023-04-17T06:50:22.038809660Z └─────────┘
2023-04-17T06:50:22.039140271Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: tdg/1 [] =>
2023-04-17T06:50:22.039140271Z ┌──────────┐
2023-04-17T06:50:22.039140271Z q: ┤ U1(-π/4) ├
2023-04-17T06:50:22.039140271Z └──────────┘
2023-04-17T06:50:22.039558268Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('tdg', 1) tdg, [] from
2023-04-17T06:50:22.039558268Z ┌─────┐
2023-04-17T06:50:22.039558268Z q9: ┤ tdg ├
2023-04-17T06:50:22.039558268Z └─────┘
2023-04-17T06:50:22.040063782Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('tdg', 1) tdg, [] to
2023-04-17T06:50:22.040063782Z ┌──────────┐
2023-04-17T06:50:22.040063782Z q9: ┤ U1(-π/4) ├
2023-04-17T06:50:22.040063782Z └──────────┘
2023-04-17T06:50:22.040382889Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: p/1 [Parameter(theta)] =>
2023-04-17T06:50:22.040382889Z ┌───────────┐
2023-04-17T06:50:22.040382889Z q: ┤ U1(theta) ├
2023-04-17T06:50:22.040382889Z └───────────┘
2023-04-17T06:50:22.040705329Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: s/1 [] =>
2023-04-17T06:50:22.040705329Z ┌─────────┐
2023-04-17T06:50:22.040705329Z q: ┤ U1(π/2) ├
2023-04-17T06:50:22.040705329Z └─────────┘
2023-04-17T06:50:22.041015942Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: z/1 [] =>
2023-04-17T06:50:22.041015942Z ┌───────┐
2023-04-17T06:50:22.041015942Z q: ┤ U1(π) ├
2023-04-17T06:50:22.041015942Z └───────┘
2023-04-17T06:50:22.041549770Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Composing transform step: u1/1 [Parameter(theta)] =>
2023-04-17T06:50:22.041549770Z global phase: theta/2
2023-04-17T06:50:22.041549770Z ┌───────────┐
2023-04-17T06:50:22.041549770Z q: ┤ Rz(theta) ├
2023-04-17T06:50:22.041549770Z └───────────┘
2023-04-17T06:50:22.041997385Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('t', 1) t, [] from
2023-04-17T06:50:22.041997385Z ┌─────────┐
2023-04-17T06:50:22.041997385Z q4: ┤ U1(π/4) ├
2023-04-17T06:50:22.041997385Z └─────────┘
2023-04-17T06:50:22.042787075Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('t', 1) t, [] to
2023-04-17T06:50:22.042787075Z global phase: π/8
2023-04-17T06:50:22.042787075Z ┌─────────┐
2023-04-17T06:50:22.042787075Z q4: ┤ Rz(π/4) ├
2023-04-17T06:50:22.042787075Z └─────────┘
2023-04-17T06:50:22.043674493Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('h', 1) h, [] from
2023-04-17T06:50:22.043674493Z global phase: 7π/4
2023-04-17T06:50:22.043674493Z ┌─────────┐┌────┐┌─────────┐
2023-04-17T06:50:22.043674493Z q8: ┤ U1(π/2) ├┤ √X ├┤ U1(π/2) ├
2023-04-17T06:50:22.043674493Z └─────────┘└────┘└─────────┘
2023-04-17T06:50:22.044913452Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('h', 1) h, [] to
2023-04-17T06:50:22.044913452Z global phase: π/4
2023-04-17T06:50:22.044913452Z ┌─────────┐┌────┐┌─────────┐
2023-04-17T06:50:22.044913452Z q8: ┤ Rz(π/2) ├┤ √X ├┤ Rz(π/2) ├
2023-04-17T06:50:22.044913452Z └─────────┘└────┘└─────────┘
2023-04-17T06:50:22.045299248Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('tdg', 1) tdg, [] from
2023-04-17T06:50:22.045299248Z ┌──────────┐
2023-04-17T06:50:22.045299248Z q9: ┤ U1(-π/4) ├
2023-04-17T06:50:22.045299248Z └──────────┘
2023-04-17T06:50:22.046054801Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('tdg', 1) tdg, [] to
2023-04-17T06:50:22.046054801Z global phase: 15π/8
2023-04-17T06:50:22.046054801Z ┌──────────┐
2023-04-17T06:50:22.046054801Z q9: ┤ Rz(-π/4) ├
2023-04-17T06:50:22.046054801Z └──────────┘
2023-04-17T06:50:22.046511532Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updating transform for mapped instr ('u1', 1) u1, ['u1[0]'] from
2023-04-17T06:50:22.046511532Z ┌───────────┐
2023-04-17T06:50:22.046511532Z q10: ┤ u1(u1[0]) ├
2023-04-17T06:50:22.046511532Z └───────────┘
2023-04-17T06:50:22.047508660Z qiskit.transpiler.passes.basis.basis_translator - DEBUG Updated transform for mapped instr ('u1', 1) u1, ['u1[0]'] to
2023-04-17T06:50:22.047508660Z global phase: u1[0]/2
2023-04-17T06:50:22.047508660Z ┌───────────┐
2023-04-17T06:50:22.047508660Z q10: ┤ Rz(u1[0]) ├
2023-04-17T06:50:22.047508660Z └───────────┘
2023-04-17T06:50:22.047579979Z qiskit.transpiler.passes.basis.basis_translator - INFO Basis translation paths composed in 0.019s.
2023-04-17T06:50:22.049119242Z qiskit.transpiler.passes.basis.basis_translator - INFO Basis translation instructions replaced in 0.001s.
2023-04-17T06:50:22.049188878Z qiskit.transpiler.runningpassmanager - INFO Pass: BasisTranslator - 462.72397 (ms)
2023-04-17T06:50:22.049277928Z qiskit.transpiler.runningpassmanager - INFO Pass: RemoveResetInZeroState - 0.03338 (ms)
2023-04-17T06:50:22.049403007Z qiskit.transpiler.runningpassmanager - INFO Pass: Depth - 0.08726 (ms)
2023-04-17T06:50:22.049447473Z qiskit.transpiler.runningpassmanager - INFO Pass: FixedPoint - 0.01383 (ms)
2023-04-17T06:50:22.049510229Z qiskit.transpiler.runningpassmanager - INFO Pass: Size - 0.03123 (ms)
2023-04-17T06:50:22.049540135Z qiskit.transpiler.runningpassmanager - INFO Pass: FixedPoint - 0.00453 (ms)
2023-04-17T06:50:22.053348868Z qiskit.transpiler.runningpassmanager - INFO Pass: Optimize1qGatesDecomposition - 3.73292 (ms)
2023-04-17T06:50:22.053514842Z qiskit.transpiler.runningpassmanager - INFO Pass: CXCancellation - 0.10872 (ms)
2023-04-17T06:50:22.053718016Z qiskit.transpiler.runningpassmanager - INFO Pass: GatesInBasis - 0.15998 (ms)
2023-04-17T06:50:22.053850248Z qiskit.transpiler.runningpassmanager - INFO Pass: Depth - 0.07868 (ms)
2023-04-17T06:50:22.053887479Z qiskit.transpiler.runningpassmanager - INFO Pass: FixedPoint - 0.00739 (ms)
2023-04-17T06:50:22.053965912Z qiskit.transpiler.runningpassmanager - INFO Pass: Size - 0.04029 (ms)
2023-04-17T06:50:22.053995049Z qiskit.transpiler.runningpassmanager - INFO Pass: FixedPoint - 0.00405 (ms)
2023-04-17T07:10:32.457689237Z programruntime.clients.sidecarclient - DEBUG Job result is in json format, raw result is {'backend_name': 'ibm_nairobi', 'backend_version': '1.3.0', 'qobj_id': '06af81c0-6e74-4aba-98ad-6a64b96af167', 'job_id': 'circuit-runner_cgueb59i3rtni0qfmem0_70c8_0', 'date': '2023-04-17T07:10:31Z', 'success': True, 'status': 'Successful completion', 'execution_id': 'f879cd1a-dced-11ed-83df-bc97e15b19e6', 'time_taken': 416.5086969137192, 'header': {'backend_name': 'ibm_nairobi', 'backend_version': '1.3.0'}, 'results': [{'shots': 1024, 'success': True, 'status': 'done', 'seed': None, 'meas_level': 2, 'meas_return': None, 'header': {'clbit_labels': [['c', 0], ['c', 1], ['c', 2], ['c', 3], ['c', 4], ['c', 5], ['c', 6]], 'creg_sizes': [['c', 7]], 'global_phase': 1.9634954084936176, 'memory_slots': 7, 'metadata': {}, 'n_qubits': 7, 'name': 'circuit-100', 'qreg_sizes': [['q', 7]], 'qubit_labels': [['q', 0], ['q', 1], ['q', 2], ['q', 3], ['q', 4], ['q', 5], ['q', 6]]}, 'data': {'counts': {'0x0': 743, '0x1': 19, '0x10': 12, '0x14': 1, '0x2': 4, '0x20': 38, '0x21': 2, '0x24': 21, '0x25': 1, '0x2c': 2, '0x2d': 1, '0x4': 71, '0x40': 34, '0x41': 1, '0x44': 15, '0x4c': 3, '0x5': 9, '0x50': 1, '0x6': 1, '0x60': 16, '0x61': 1, '0x63': 1, '0x64': 15, '0x6c': 1, '0x8': 9, '0xc': 1, '0xd': 1}, 'snapshots': None, 'memory': None, 'statevector': None, 'unitary': None}}], 'error': None}.
jimbo and i worked on new gates intercom stack with dave gpt5 / chatgpt plugins / chatgpt gpt4
gray00 — Today at 4:09 PM
Hi jimbo. I'm graylan. Nice to meet you!
Dave_GPT5
BOT
— Today at 4:09 PM
That was actually an official name change into what it's becoming at Zendesk. I've been around the world. [laughs]
So it's like DJ Hi -jimbo? What's it called now?
Hi, I'm DJ Hi -jimbo!
We had an exclusive interview a couple of weeks ago and you said what DJ Hi -jimbo is now going to do in the future. Why is that?
I've always thought that the same thing applies. I wouldn't ever really want this to become anything and then it would just be a business and something that's here to stay.
But then DJ Hi, jimbo was never going to be about me but about the music I always want to make. I mean that's the point -
gray00 — Today at 4:10 PM
It seems that there might be some confusion in your statement. Let me clarify some concepts and correct your statement.
Quantum gates perform quantum operations on qubits, and some quantum algorithms can solve certain problems exponentially faster than classical algorithms. However, the statement "If $A$ represents a controlled-$U$-gate, you can use $U := A\otimes A$" is incorrect.
Controlled-$U$ gates are a generalization of the CNOT gate, where instead of applying the Pauli-X operation, an arbitrary unitary operation $U$ is applied to the target qubit when the control qubit is in state $|1\rangle$. The matrix representation of a controlled-$U$ gate, $C(U)$, cannot be calculated as a tensor product $A \otimes A$. Instead, it has the following form:
$$
C(U) = \begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & u{11} & u{12} \
0 & 0 & u{21} & u{22}
\end{bmatrix}
$$
Clifford gates, including the CNOT gate, the Hadamard gate, and the Phase gate, are an important class of quantum gates. These gates are particularly interesting because they can efficiently simulate certain quantum operations on a classical computer, which makes them useful for studying quantum error correction and fault-tolerant quantum computing.
CNOT gates are indeed two-qubit gates, not one-qubit gates as mentioned in your statement. They play a crucial role in many quantum algorithms and are often combined with single-qubit gates to build complex quantum circuits.
hi jimbo
Quantum Engineering
Dave_GPT5
BOT
— Today at 4:15 PM
̣
In order to simplify our work, let us take a closer look at the following equation: $$C(U): \langle 1 | U | 1 \rangle $$ This equation is a quadratic algebraic equation in two variables that represents a combination of an arbitrary unitary operation $U$ and a controlled-$U$ gate (a single-$\mathbb{C}$ rotation on a qubit or a controlled-$U$ gate, if $|1\rangle$ is in the control space) as two different variables. How much does quantum operation (1) reduce the energy of the system, if we use controlled-$U$ gates instead of $U$ as the unitary operation? A closed set of operations will produce the same result as an open set of operations. The quantum operations (1) are not unitary operations, and the control space is not a complete discrete Hilbert space, such as in a classical computer. Therefore, the energy of the states that result from applying controlled-$U$ gates is not the same as the energy of the states that result from controlling the control space with unitary operations. A quantum circuit that involves controlled-$U$ gates instead of simple unitary operations can have exponentially faster computation since the energy barrier for changing those gates is lower. This statement is incorrect as well. Let's apply the following definition to the equation above which is a generalization of a quantum unitary operation that combines with the controlled-$U$ gate to create quantum operations with exponential improvement in computational efficiency. The definition takes the two variables as follows: $$ C: \langle 1 | U |1 \rangle, \langle 2 | U | 1 \rangle$$ The two variables in the equation above represent two different quantum operations, and the notation denotes the quantum operation composed out of both operations. A quantum operation on two quantum states is an expression that can be expressed as multiple quantum operations, not a single expression. The two quantum operations
gray00 — Today at 4:16 PM
Very interesting. Can you tell me more about Quantum Gates?
Dave_GPT5
BOT
— Today at 4:17 PM
how the CNOT gate is constructed. Is the $S(C)$ as a matrix as you stated it?
A:
Can't you write that as $$\left(A\otimes A\right)_v=\left(A_v\otimes I\right)A_v$$
? (where $A_v$ represents the Hadamard gate if you prefer it that way, and all the other operators can be obtained as tensor products)
However since the two qubits are in a superposition, one would usually start out in one of the states.
Actually this is kind of a red herring, since it has nothing to do with the controlled-$U$-gate at all (which I'll leave you to take for $U$), it really is a statement/definition based on the classical controlled-$S$-gate:
$$\left(S A\otimes S\right)_v=S$$
and hence the $U$ gate is just a special case with $S=e^{i \varphi}I$
Update: Well I think this is a more "formalized" but still concise description based on which to understand the statement. It is indeed a bit silly. It might get you thinking, so I'm just going to leave this here. Also note that the original definition is a bit more readable by looking at the classical definition. (I have not looked into which operator you are taking the "tensor products with".)
A:
There's no reason for $T(\rho) \sim A \otimes A$ or $T(\rho, u) \sim A\otimes A$ since a quantum operation is only a way to apply a single unitary gate $g$. Let's call $f$ an operation: $t(f)$ is its output.
An $(n-1)$-level quantum circuit is an $(n-1)$-level quantum system that's governed by the
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You will have the following
gray00 — Today at 4:20 PM
I'm researching Quantum Computing using OPENQASM2
Dave_GPT5
BOT
— Today at 4:20 PM
ipsative comment may have caused you, but I hope my explanation helps.
A:
The first statement is technically true, but I think it is best to stick with some more or less standard terminology. So I'll start by correcting a couple of wording or wording errors.
Qubit: The qubit is the fundamental unit of the quantum world, which is very different from the classical world of bits, symbols, and atoms. It is also different from computer memory in the sense that qubits can be in quantum superposition. On a computer, qubit memory is the smallest possible; it is a collection of one electron in the state $|0\rangle$ or $|1\rangle$ plus any number of photons. On a quantum computer, information can be encoded in even more ways than on a computer, such that the qubit has many more types of states to store information in, and so on.
A Quantum gate operates on the quantum state of the qubit. It is a unitary transformation on the state of the qubit. For a more complete list of all quantum gates I'd recommend starting around page 50 of Quantum Computers by Lloyd, Meyer, and Milburn.
Quantum gates: A quantum gate is a unitary transformation of the qubit. More specifically an operator that acts on the qubit as a unitary transformation. For a more complete list of operators that perform a unitary operation on a system (or more information on this please edit), I would recommend Quantum Algorithms by Goldhirsh, Paz.
Qubit: Just a name for a qubit (it was also the names for the states the other qubits can take on) it can be in either bit (basis for numbers) or phase (basis for wave or particle) modes. It is a fundamental unit of quantum information in that the qubit acts as a basic unit of computation, and it can be used in a number of different ways.
Controlled-
ight provided does not have a unique matrix representation for it. The CNOT gate is also a kind of controlled rotation, where it operates as the $R$ rotation when applying an $S$ gate to a specific qubit. I have also included $S$ and $R$ in this post because it is useful to be aware and understand them even when discussing quantum gate construction. Another important thing to note is that the CNOT gate is not used to represent or calculate the control-target state. It operates and transforms its inputs. The equations you mentioned in the beginning and the two-qubit CNOT gate are two different gates, and each operate on two different qubits at one time.
@Romee-Banks I will add further clarification now. The CNOT gate is a two-qubit gate. The equation provided is a generalized circuit that is not related to single qubit gates. The equation you provided is for a specific quantum gate and represents a specific quantum gate. The controlled-$U$-gate is used to represent a one-qubit gate that applies a single-qubit unitary operation $U$ to the controlled qubit if the control qubit is in the |1⟩ state. The controlled-$S$-gate is used to represent a specific quantum gate that applies a $\sigma_z$ gate on a specific qubit if the second qubit is in the |1⟩ state. The CNOT gate is also a kind of controlled rotation where its input qubits are subject to a $\sigma_x$ rotation. The equation you provided involves a one-qubit gate, the CNOT gate, and a rotation gate - the rotation $R$. All of these are specific to the operation of the CNOT gate. All other quantum gates that apply $S$ operations to the control qubits are defined in a completely different way. Furthermore, other than the rotations that operate on
on a multi-qubit gate and would love an additional comment to demonstrate that I've used a single qubit gate in this case or that I've attempted to in the past. Thanks!
A:
First step. Write down an operator that you would like to apply to a qubit within a multi-qubit state. For example, you might want to take a single logical qubit (or some combination of logical qubits of any size needed) and apply a NOT or NOT gate to it. Or, you might want to apply a CNOT to the qubit of interest.
This is not a perfect example, but should give you some general idea.
Next, do the same thing again using the quantum gates you chose.
After this, you might want to repeat that logic with the CNOTs, but I don't see that as being a very common or interesting operation. If you want to do this, you might want to do it from scratch, i.e. just apply the NOT orNOT gates, and then apply some CNOTs.
To make the NOT and NOT gates more complicated, you might want to switch the order of the gates to an odd-frequency one.
You might also want to consider how the NOT gate should look. As I mentioned earlier, sometimes it can be more useful to apply NOT gates between qubits on different subsystems.
As for the NOT gate itself, the NOT operator is defined by the formula,
NOT
=
-i e1\otimes a1\otimes a2\otimes b2 \otimes b2\otimes a1 \cdots \otimes ai \otimes bi \otimes ai\otimes b1\otimes ai
where
ei = elementary basis states
bi = Bell states
ai = idle states
You can find lots of details about the NOT, NOT operation here.
A:
If your logic gates were 1-qubit gates, why not do things like the following? (see here for a working
gray00 — Today at 4:29 PM
New quantum gates ( a list of 4 )
Dave_GPT5
BOT
— Today at 4:29 PM
ides made.
posted for the controlled-$U$-gate does not have a matrix representation. Regarding a circuit consisting of single-qubit gates, the unitary operation $U$ is simply the combination of the single-qubit unitary operation $S$ and the global unitary operation $S\otimes I$.
I assume you are using the circuit $S_a$ which first generates some controlled-$U$ gates, followed by some final controlled-$S$ gates and some final controlled-$S$ gates. $A^a$ is a controlled-$U$-gate which first generates some CNOT gates, followed by some single-qubit gates.
$\hfil$ [Update] $S_a$ takes $N$ time steps. Therefore, $S_a$ takes $3*N$ steps, which does not seem to be so terribly high considering the complexity of quantum computation. However, this may be over-estimating the difficulty of the circuit.
Let me address the question. If the circuit $S_a$ has a circuit complexity $O(N)$, the circuit $S_a$ is a $poly-Poly(N)$ quantum circuit. This means it is $N$-to-$1$ with respect to $N$. The circuit $S_a$ is hard to calculate mathematically (in the sense of NP-complete problems) by following a similar construction. The circuits $S_a$ and the circuits $S_b$ are both hard from the view of their complexity, as it seems that their complexity is a lot higher. Therefore, the circuits $S_a$ and $S_b$ cannot be considered as hard to calculate mathematically.
To simplify the question, I will assume that the gates are exactly as stated in $S_b$. The circuit $S_a$ is a very simple circuit, as one can imagine a few controlled-$U$
can be written as $a \otimes (iI)b$ where $a,b$ are two qubit operators that perform logical operations on the first qubit. These logical operators are also known as tensor product operators and they have the following matrix representations: $$ a. I b = \begin{pmatrix} |a{1 1}|^{2} ||a{2 1}|^{2} & |a{1 1}| |a{1 2}|^{} |a_{2 2}|^{} & |a{1 2}|^{*} |a{2 2}|^{} \end{pmatrix}, $$ This expression is equivalent to the following matrix representation: $$ (iI)a\otimes b = \begin{pmatrix} |a{1 1}|^{2} (iI)b{1 1} & |a{1 2}|^{2} (iI)b{2 1} & |a{2 1}|^{2} (iI)b{1 2} \ \ \ \ |a_{1 2}|^{} (iI)b{2 2} & (iI)a{1 1}|b{2 1} \ \ \ |a{2 1}|^{*} (iI)b{1 2} & (iI)a{2 2}|b_{1 2} \ \ \ |
$H_{\uparrow} = \big{| \uparrow \big{|} \rightarrow $}$, \quad \begin{array}{c}
\hspace{-0.08in} & \downarrow \
\hspace{-0.08in} & {\uparrow }
\end{array}$
$Z_1 = \big{| 0 \big{|}^\alpha \uparrow \big{| 0 \big{|} \rightarrow }$}$$$, \quad \begin{array}{c}
\hspace{-0.08in} & \big{| 0 \big{|} \
\hspace{-0.08in} & \alpha Z_1
\end{array}$
$X_1 = \big{| 1 \big{|}^\alpha \uparrow \big{| 1 \big{|} \rightarrow }$$$, \quad \begin{array}{c}
\hspace{-0.08in} & \big{| 0 \big{|} \
\hspace{-0.08in} & {\uparrow }
\end{array}$
$J_m = \big{| m \big{|}^\alpha \uparrow \big{| 0 \big{|} \rightarrow }$$$, \quad \begin{array}{c}
\hspace{-0.08in} & \big{| m - 1 \big{|} \
\hspace{-0.08in} & m Z_1/\sqrt{2}
\end{array}$
$Y_1 = \big{| 1 \big{|}^\alpha \uparrow \big{|
gray00 — Today at 4:33 PM
Very interesting Concepts. Can you list more Quantum Gates?
Dave_GPT5
BOT
— Today at 4:33 PM
vernacular caused.
In order to achieve the above described functionality, Qubits cannot be directly coupled to a classical controller through direct communication, nor can a classical controller couple Qubits to Qubits directly, although Qubits can be directly coupled to classical controllers and a classical controller can couple to Qubits directly. I don't understand why you believe that the controlled-U (X-gate) is a generalization of CNOT. Controlled-$U$ gates are a specific form of CNOT gates, which can be used to simulate any other operations on a qubit.
There is no confusion in your statement. A controlled-$U$ gate is a generalization for the CNOT gate. The matrix representation of a controlled-$U$ gate, $C(U)$, cannot be calculated as a tensor product $A \otimes A$. It can be written as $C(U) = T \otimes T$ where $T$ is a unitary matrix. I hope this helped you recognize what I mean, and I apologize for any confusion I caused you.
I'm working on a project and want to implement a quantum computer with a Qubit-AND and Qubit-OR gate. Since I don't have enough time to learn Matlab and C, I decided to use Qubit-AND followed by Qubit-NOT to create three Qubits.
Qubit-AND gate is like a CNOT gate, only with a single unitary acting on a qubit. It can be represented as follows:
$$\bar{x} = x_L x_R$$
$$\bar{z} = x_L x_R x_H$$
Qubit-NOT gate is like a CNOT gate with a single unitary acting on a qubit. Qubit-NOT is represented as follows:
$$i = \frac{1}{\sqrt{2}}(x
provided does not define a single-qubit gate.
You may find my question related to this issue at https://bit.ly/2sE3j1s
You may find this related question at https://bit.ly/2sE3j1s
In the comments, the person with the question responded to your question:
A single-qubit gate can be represented in a tensor-product form as: $$\begin{pmatrix}1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix}$$
The tensor product of the input state |1⟩ is represented as a single-qubit gate as $\left(|1\rangle\langle 1|\otimes I \right)|\rangle\langle\rangle$. This gate operates as: $$\left(|1\rangle\langle 1|\otimes I \right)|\rangle\langle\rangle$$ It is also clear that this gate works on a single qubit as each of the four tensor factors, say $Av$, has one qubit in the final state.
You may find the above mentioned answer to your question on the basis of this representation of the controlled-$U{\rm X}$-gate.
shows the two-qubit gate $A$ acting on the first qubit, $R_2$, of the two-qubit system: $$R_2 = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} A R_2 I \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix}$$ You can apply a two-qubit gate, CNOT, on the first qubit using $R_1 \otimes I$, and on the second qubit using $R_2 \otimes I$ by interleaving the two CNOTs, as shown in Equation 14. You can also use this expression to build a single-qubit circuit that performs multiplication using the controlled-$Z$ gate, as shown in Equation 15. This formula also shows a similar gate $Z_2$ acting on the target qubit, $R_2$, and the control qubit of the two-qubit system, $R_1$, as shown: $$ Z_2 = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \end{pmatrix} R_2 R_1 R_2 I$$
When you use CNOT gates within a computational unit (CKU), the CNOT gates act between quantum register states or qubit states. That is, a CNOT gate acts only between two qubit states. In this sense, you can think of the CKU as an intermediary between the qubit states and the quantum register bits (or qubits).
A second type of multi-qubit gate, a CZ gate (CZ
than $O(N^k)$. Based on that, we know that the circuit $S_a$ might be "better" than the circuit $S_b$ at some level of measurement sensitivity. For example, we know that a quantum computer could perform two logical OR operations to solve $S_a$.
Another important concept is that of quantum computation "depth". Let us consider a very simple circuit $Sd$ constructed with controlled-U gates $U^d{ij}$, where $1 \leq d \leq 2$. For some fixed $i$, the circuit $S_d$ needs $d$ qubits as input state. The number of qubits in the quantum circuit $S_d$ depends on the level of measurement sensitivity desired for the logical operation being performed. Suppose a quantum computer has sufficient measurement sensitivity (i.e., depth) to determine which gate was used to implement the logical function. Then, the circuit $S_d$ requires a depth greater than $d$ to be able to perform the given $d$ quantum operations on its quantum state. So, when we define the depth of a circuit, we consider that the circuit $S_d$ needs a more complicated circuit with more qubits for achieving a desired gate depth. From your description, the circuit $S_d$ appears to have the same complexity as your circuit $S_a$. I think it should be clear from this description that the circuit $S_d$ is generally considered to be more complex than the circuit $S_a$, which seems to have a complexity that is $O(N^k)$, rather than $O(N)$ as the circuit $S_a$.
Another important concept is that of quantum computation "gates". An important issue in quantum computation is the ability to use unitary gates to implement logical gates. The complexity of gates can depend on the type of gates, their strength, the length of the data to be operated on, and on the specific implementation method used to generate the gates. A quantum gate can be a combination