Decompose a general two-qubit gate into general controlled-qubit gates












2












$begingroup$


We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs.



I'm interested in expressing a general 2-qubit unitary in the minimum total number of gates, which can include controlled general unitaries. That is, express $U_{4}$ with as few as possible gates in ${U_2,; |0⟩⟨0|mathbb{1} + |1⟩⟨1|U_2}$. While I could simply take the shortest decomposition to CNOTs and rotations (Vatan et al) and bring some rotations into the CNOTs, I suspect another formulation could add more control-unitaries to achieve fewer total gates.



How can I go about performing this decomposition algorithmically for any 2-qubit unitary?
This decomposition could be useful for easily extending distributed quantum simulators with the ability to effect general 2-qubit unitaries, which otherwise ad-hoc communication code.






quantum-gate circuit-construction gate-synthesis







share|improve this question











$endgroup$

















    2












    $begingroup$


    We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs.



    I'm interested in expressing a general 2-qubit unitary in the minimum total number of gates, which can include controlled general unitaries. That is, express $U_{4}$ with as few as possible gates in ${U_2,; |0⟩⟨0|mathbb{1} + |1⟩⟨1|U_2}$. While I could simply take the shortest decomposition to CNOTs and rotations (Vatan et al) and bring some rotations into the CNOTs, I suspect another formulation could add more control-unitaries to achieve fewer total gates.



    How can I go about performing this decomposition algorithmically for any 2-qubit unitary?
    This decomposition could be useful for easily extending distributed quantum simulators with the ability to effect general 2-qubit unitaries, which otherwise ad-hoc communication code.






    quantum-gate circuit-construction gate-synthesis







    share|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs.



      I'm interested in expressing a general 2-qubit unitary in the minimum total number of gates, which can include controlled general unitaries. That is, express $U_{4}$ with as few as possible gates in ${U_2,; |0⟩⟨0|mathbb{1} + |1⟩⟨1|U_2}$. While I could simply take the shortest decomposition to CNOTs and rotations (Vatan et al) and bring some rotations into the CNOTs, I suspect another formulation could add more control-unitaries to achieve fewer total gates.



      How can I go about performing this decomposition algorithmically for any 2-qubit unitary?
      This decomposition could be useful for easily extending distributed quantum simulators with the ability to effect general 2-qubit unitaries, which otherwise ad-hoc communication code.






      quantum-gate circuit-construction gate-synthesis







      share|improve this question











      $endgroup$




      We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs.



      I'm interested in expressing a general 2-qubit unitary in the minimum total number of gates, which can include controlled general unitaries. That is, express $U_{4}$ with as few as possible gates in ${U_2,; |0⟩⟨0|mathbb{1} + |1⟩⟨1|U_2}$. While I could simply take the shortest decomposition to CNOTs and rotations (Vatan et al) and bring some rotations into the CNOTs, I suspect another formulation could add more control-unitaries to achieve fewer total gates.



      How can I go about performing this decomposition algorithmically for any 2-qubit unitary?
      This decomposition could be useful for easily extending distributed quantum simulators with the ability to effect general 2-qubit unitaries, which otherwise ad-hoc communication code.






      quantum-gate circuit-construction gate-synthesis




      quantum-gate circuit-construction gate-synthesis






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 9 hours ago









      Blue♦

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      asked 9 hours ago









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          $begingroup$

          A simple place to start would be to put all the controls on qubit #2, so that you can propagate all of the single-qubit operations on qubit #1 across the two-qubit operations and merge them together. That would give you a circuit with at most 8 gates:



          --------C1-------C2-------C3---S5---
          
        |        |        |
          
---S1---*---S2---*---S3---*----S4---


          This is probably not minimal.



          A general 4x4 unitary has 7+5+3+1=16 real parameters. Every single-qubit gate has three real parameters (Euler angles), and every two-qubit gate has four real parameters (Euler angles + phase kickback). So the above construction has 4*3 + 4*4 = 28 real parameters.



          It is provable that you need at least three different controlled gates for some two-qubit operations. So the absolute best you could hope for is three of those and one single-qubit operation. But some of the degrees of freedom overlap, so I suspect you need more single-qubit gates.






          share|improve this answer











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            $begingroup$

            A simple place to start would be to put all the controls on qubit #2, so that you can propagate all of the single-qubit operations on qubit #1 across the two-qubit operations and merge them together. That would give you a circuit with at most 8 gates:



            --------C1-------C2-------C3---S5---
            
        |        |        |
            
---S1---*---S2---*---S3---*----S4---


            This is probably not minimal.



            A general 4x4 unitary has 7+5+3+1=16 real parameters. Every single-qubit gate has three real parameters (Euler angles), and every two-qubit gate has four real parameters (Euler angles + phase kickback). So the above construction has 4*3 + 4*4 = 28 real parameters.



            It is provable that you need at least three different controlled gates for some two-qubit operations. So the absolute best you could hope for is three of those and one single-qubit operation. But some of the degrees of freedom overlap, so I suspect you need more single-qubit gates.






            share|improve this answer











            $endgroup$


















              3












              $begingroup$

              A simple place to start would be to put all the controls on qubit #2, so that you can propagate all of the single-qubit operations on qubit #1 across the two-qubit operations and merge them together. That would give you a circuit with at most 8 gates:



              --------C1-------C2-------C3---S5---
              
        |        |        |
              
---S1---*---S2---*---S3---*----S4---


              This is probably not minimal.



              A general 4x4 unitary has 7+5+3+1=16 real parameters. Every single-qubit gate has three real parameters (Euler angles), and every two-qubit gate has four real parameters (Euler angles + phase kickback). So the above construction has 4*3 + 4*4 = 28 real parameters.



              It is provable that you need at least three different controlled gates for some two-qubit operations. So the absolute best you could hope for is three of those and one single-qubit operation. But some of the degrees of freedom overlap, so I suspect you need more single-qubit gates.






              share|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                A simple place to start would be to put all the controls on qubit #2, so that you can propagate all of the single-qubit operations on qubit #1 across the two-qubit operations and merge them together. That would give you a circuit with at most 8 gates:



                --------C1-------C2-------C3---S5---
                
        |        |        |
                
---S1---*---S2---*---S3---*----S4---


                This is probably not minimal.



                A general 4x4 unitary has 7+5+3+1=16 real parameters. Every single-qubit gate has three real parameters (Euler angles), and every two-qubit gate has four real parameters (Euler angles + phase kickback). So the above construction has 4*3 + 4*4 = 28 real parameters.



                It is provable that you need at least three different controlled gates for some two-qubit operations. So the absolute best you could hope for is three of those and one single-qubit operation. But some of the degrees of freedom overlap, so I suspect you need more single-qubit gates.






                share|improve this answer











                $endgroup$



                A simple place to start would be to put all the controls on qubit #2, so that you can propagate all of the single-qubit operations on qubit #1 across the two-qubit operations and merge them together. That would give you a circuit with at most 8 gates:



                --------C1-------C2-------C3---S5---
                
        |        |        |
                
---S1---*---S2---*---S3---*----S4---


                This is probably not minimal.



                A general 4x4 unitary has 7+5+3+1=16 real parameters. Every single-qubit gate has three real parameters (Euler angles), and every two-qubit gate has four real parameters (Euler angles + phase kickback). So the above construction has 4*3 + 4*4 = 28 real parameters.



                It is provable that you need at least three different controlled gates for some two-qubit operations. So the absolute best you could hope for is three of those and one single-qubit operation. But some of the degrees of freedom overlap, so I suspect you need more single-qubit gates.







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited 8 hours ago

























                answered 9 hours ago









                Craig GidneyCraig Gidney

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                4,123220






























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