Control logic decompositions

Controlled quantum gates are a fundamental building block for algorithm design and unitary synthesis. Essential techniques like linear combinations of unitaries, Select-applying a gate (qml.Select), or QROMs (demo) all use controlled gates.

Consequently, decomposing (groups of) controlled quantum gates is an important step in quantum compilation.

Inputs

• Circuit description containing quantum control logic.

Outputs

• Circuit description containing reduced quantum control logic. The reduction is typically seen in the number of control nodes and/or the number of controlled gates.

Covered decompositions

We show the following four decompositions:

1. Control of a compute/uncompute pattern [1]
2. Linear combination of QSVTs
3. Controlled global phases [2]
4. Flipped and controlled phase shifts

Examples

We give one brief example for each of the four techniques. See the respective tabs for more details and examples.

• Controlled versions of a compute/uncompute pattern can be simplified. This allows us to skip control nodes like in the following example:

     0: ─╭RX(0.50)─╭●────╭●─╭RX(0.50)†─┤  =     0: ──RX(0.50)─╭●────╭●──RX(0.50)†─┤
     1: ─│─────────├X─╭●─├X─│──────────┤        1: ───────────╰X─╭●─╰X────────────┤
     2: ─│─────────│──├X─│──│──────────┤        2: ──────────────├X───────────────┤
  ctrl: ─╰●────────╰●─╰●─╰●─╰●─────────┤     ctrl: ──────────────╰●───────────────┤.
  

• Linear combinations of QSVT circuits can be condensed, leading to simplifications like the following:

  aux: ──H─╭○─────╭○────────────╭○─────╭○─────────────╭○──────╭●─────╭●────────────╭●─────H─┤↗₀│
  c:   ────╰∏_ϕ_0─├BlockEncode0─╰∏_ϕ_1─├BlockEncode0†─╰∏_ϕ_2──╰∏_τ_0─├BlockEncode0─╰∏_τ_1─────────┤
  t:   ───────────╰BlockEncode0────────╰BlockEncode0†────────────────╰BlockEncode0────────────────┤.
  
  =
  
  aux: ──H─────╭●─────────────────────────╭●─────╭○─────────────╭○─────H─┤↗₀│
  c:   ──∏_ϕ_0─╰∏_θ_0─╭BlockEncode0─∏_ϕ_1─╰∏_θ_1─├BlockEncode0†─╰∏_ϕ_2────────┤
  t:   ───────────────╰BlockEncode0──────────────╰BlockEncode0†───────────────┤.
  

• Controlled global phases, unlike their non-controlled counterparts, cannot simply be skipped. Instead, they decompose into phase shift gates, with any of the controls acting as new target:

  0: ──╭GlobalPhase(a)─┤  =  0: ──────────┤  =  0: ──────────┤  =  0: ──────────┤
  1: ──├●──────────────┤     1: ──╭Rϕ(-a)─┤     1: ──╭●──────┤     1: ──╭●──────┤
  2: ──├●──────────────┤     2: ──├●──────┤     2: ──├Rϕ(-a)─┤     2: ──├●──────┤
  3: ──╰●──────────────┤     3: ──╰●──────┤     3: ──╰●──────┤     3: ──╰Rϕ(-a)─┤
  

• (Controlled) phase shift gates, in turn, can be decomposed via this relationship into (controlled) $R_Z$ gates, leading to staircase circuits:

  0: ──╭Rϕ(a)─┤  ≅  0: ──╭RZ(a)───────────────────┤
  1: ──├●─────┤     1: ──├●─────╭RZ(a/2)──────────┤
  2: ──╰●─────┤     2: ──╰●─────╰●────────RZ(a/4)─┤
  

Typical usage

These decomposition tricks are usually applied to high-level or intermediate-level circuit descriptions that allow insight into the control logic applied to some subroutine.

References

[1] "A software methodology for compiling quantum programs", Thomas Häner, Damian S. Steiger, Krysta Svore, Matthias Troyer, Quantum Sci. Technol. 3 020501, 2018

[2] "Quantum computation and quantum information", Michael A. Nielsen, Isaac L. Chuang, Cambridge Press, 2010, Cambridge University Press