Table 1.
Summary of our results and comparison with previous work. Note that we report here worst-case values, corresponding to k ∼ sn and for and , respectively; see the respective sections for more comprehensive discussions.
| Dicke state | Reference | Depth | Ancillas [Dimension] | Repetitions |
|---|---|---|---|---|
| SU(2) spin-s | NRR [22] | 𝒪(skn) | 0 | 1 |
| Result 1 | 𝒪(skn) | 1 [k + 1] | 1 | |
| Result 2 | 𝒪 (log(sn)) | 𝒪(log(sn) + n) [2] | ||
| Result 3 | 𝒪(1) | 𝒪(n) [2sn + 1] | ||
| Result 4 | 𝒪(1) | 𝒪(log(sn)) [2], 𝒪(n log(sn)) [2s + 1] | ||
| SU(d) | NR [23] | 𝒪(nd) | 0 | 1 |
| LCG [9] | 𝒪(log n) | 𝒪(n log n + log d) [2] | 𝒪(1) | |
| Result 5 | 𝒪((n/d)d) | 1 [2], 1[𝒪((n/d)d)] | 1 | |
| Result 6 | 𝒪(d log n) | 𝒪(d log n + n) [2] | 𝒪(n(d−1)/2) | |
| Result 7 | 𝒪(d) | 𝒪(n + d) [n + 1] | 𝒪(n(d−1)/2) | |
| Result 8 | 𝒪(1) | 𝒪(d log n) [2], 𝒪(nd log n) [d] | 𝒪(n(d−1)/2) |
Figure 1.
Figure 2.
Circuit diagram for preparing the state sequentially (11) (a) , with x = max(0, 2s(i – n – 1) + k) and y = min(2si – 1, k – 1); (b) .
Figure 3.
Circuit diagram for preparing the state in log depth using the standard QPE algorithm. All ancilla wires are qubits. The initial state of the bottom wire is (24), and U is defined in (27).
Figure 4.
Circuit diagram for preparing the state in constant depth using the Hadamard test. The top wire is a qudit of dimension d = 2sn + 1. The initial state of the bottom wire is (24), and 𝒰 is defined in (35).
Figure 5.
Circuit diagram for preparing the state , which can be implemented in constant depth. The top ℓ wires are qubits, while all other wires are qudits of dimension 2s + 1. The state |ψ(s, p)〉 is given by (23), and U(x) is defined in (38).
Figure 6.
Circuit diagram for .
Figure 7.
Circuit diagram for preparing the state in log depth using the standard QPE algorithm. All ancilla wires are qubits. The initial state of the bottom wire is (66), and U(i) is defined in (69).
Figure 8.
Circuit diagram for preparing the state in constant depth using Hadamard tests. All ancilla wires are qudits of dimension 𝔡 = n + 1. The initial state of the bottom wire is (66), and 𝒰(i) is defined in (76).
Figure 9.
(a) Circuit diagram for preparing the state in constant depth. Each of the top (d – 1) wires represent ℓ qubits, while each of the other wires represent n qudits of dimension d. F is a fan-out gate. (b) Decomposition of the sub-circuit, where Ui(x) is defined in (79).