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Quantum-state preparation with universal gate decompositions

Martin Plesch, Časlav Brukner

arXiv:1003.5760v2quant-ph

TL;DR

Generic quantum-state preparation must balance CNOT count and circuit depth because entangling gates are experimentally demanding. The paper gives an explicit CNOT-and-rotation scheme that reduces the leading-order CNOT prefactor for even n and roughly halves depth, including improvements for four qubits.

  • Problem

    Generic quantum-state preparation requires exponentially many CNOT gates, while minimizing entangling-gate count and circuit depth remains important for experimental precision.

  • Method

    The paper gives an explicit state-preparation circuit using one-qubit rotations and CNOT gates, with phases that can exploit parallel execution.

  • Results

    23/24 2^n is the leading-order CNOT bound for even n, while four qubits require 9 rather than 11 CNOT gates.

  • Takeaways & Limitations

    The scheme is intended to help design and build small-scale quantum circuits using present technologies.

  • Takeaways & Limitations

    For odd n, the stated bound is weaker than the even-n bound, although further optimization is possible because the final operation is not a completely defined unitary.

Abstract

from arXiv · show

In quantum computation every unitary operation can be decomposed into quantum circuits-a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-NOT (CNOT) gates. Two measures are important when judging the complexity of the circuit: the total number of CNOT gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 2. It also reduces the total number of CNOT gates from 2^n to 23/24 2^n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 11 CNOT gates to 9 and the depth from 11 to 5 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.

INTRODUCTION

Quantum circuits can use arbitrary one-qubit rotations and a single entangling two-qubit gate, while state-preparation complexity is constrained by CNOT count and circuit depth. The paper targets exponential CNOT costs for generic states and reduces both the leading-order count for even n and the depth bound.

  • INTRODUCTION: Arbitrary unitaries can be decomposed into one-qubit operations and a single two-qubit gate type such as CNOT.One-qubit operations alone cannot generate general entanglement.
  • INTRODUCTION: CNOT count is experimentally important because two-qubit gates are harder to realize and introduce more imperfections than one-qubit gates.Each additional CNOT increases overall circuit imperfection.
  • INTRODUCTION: An exponential number of CNOT gates is generally required for an n-qubit unitary, motivating optimization of the gate count.The exponential dependence follows from counting the parameters of a general unitary operation.
  • INTRODUCTION: State preparation requires only a transformation from a known input state to a target state, rather than a completely specified unitary.A whole class of unitaries can satisfy the same input-to-target mapping.
  • INTRODUCTION: 23/24 2^n is the proposed leading-order CNOT upper bound for generic state preparation with even n, improving the best known prefactor c = 1.The supplied passage states the prefactor reduction and the four- and six-qubit improvements.
  • INTRODUCTION: At most half as many computation steps as CNOT gates are needed in the proposed scheme because at least two gates can run in parallel per step.Circuit depth is defined as the minimal number of computation steps required.

LOWER BOUNDS

The paper derives lower-bound constraints from the number of parameters in arbitrary pure states and from the maximum number of parallel CNOT gates per computation step. A four-qubit state illustrates the parameter accounting through Schmidt decomposition.

  • LOWER BOUNDS: A general n-qubit pure state is fully described by 2^(n+1) − 2 real parameters.Preparation introduces these parameters through one-qubit rotations separated by CNOT gates.
  • LOWER BOUNDS: Starting from a product state, local rotations contribute two parameters per qubit because the third Euler angle changes only the global phase.Thus, n qubits with k CNOT gates have fewer independent parameters than a naive three-per-qubit count suggests.
  • LOWER BOUNDS: At most n/2 CNOT gates can be performed in one computation step, so circuit depth also has an exponential lower bound with a linear correction.The depth optimization can reduce only the prefactor, up to a linear correction.
  • LOWER BOUNDS: A four-qubit state can be factorized into two two-qubit parts and represented using a Schmidt decomposition.The decomposition organizes the state into paired orthogonal states on the two halves.
  • LOWER BOUNDS: The four-qubit Schmidt parameterization contains 30 real parameters, matching the general count 2^5 − 2.The component states contribute 6, 4, 2, and 0 parameters per half, while the coefficients contribute 6.

Phase 1

Phase 1 prepares generalized Schmidt coefficients on the first two qubits from the known state |00⟩. Because the operation is state preparation rather than a fully specified unitary, it requires only one CNOT with suitable one-qubit rotations.

  • Phase 1: The first phase prepares the generalized complex Schmidt coefficients on the first two qubits, starting from |00⟩.This produces the coefficient state required by the generalized Schmidt decomposition.
  • Phase 1: One CNOT plus suitable one-qubit rotations realizes this two-qubit state-preparation operation.The operation need not implement a completely defined unitary transformation.

Phase 2

Phase 2 transfers the first two qubits’ computational-basis labels to the second two qubits using two CNOTs. This creates matching basis-state components while preserving the Schmidt coefficients.

  • Phase 2: Two CNOTs copy the computational-basis states of the first two qubits onto the corresponding states of the second two qubits.One CNOT uses the first qubit as control and third as target; the other uses the second and fourth qubits.
  • Phase 2: The resulting four-qubit state has the same Schmidt decomposition coefficients as the target state.This phase needs no one-qubit rotations.

Phase 3

Phase 3 applies a unitary operation that maps computational-basis states of the first two qubits to the four Schmidt-basis states |ψ⟩i, using a decomposition requiring three C-NOT gates.

  • Phase 3: Three C-NOT gates implement the unitary mapping the first two qubits’ computational-basis states to the states |ψ⟩i.The states |ψ⟩i arise from the Schmidt decomposition.
  • Phase 3: Phase 3 is part of a four-phase sequence for arbitrary four-qubit state preparation.The full sequence is depicted with single-qubit rotations between C-NOT gates.

Phase 4

Phase 4 transforms the third and fourth qubits’ computational-basis states into the Schmidt-basis states, using three C-NOT gates. Together, the four phases require 9 C-NOT gates and have depth 5.

  • Phase 4: Three C-NOT gates implement the unitary operation that maps the third and fourth qubits into the Schmidt basis.This is the final phase of the four-qubit preparation circuit.
  • Phase 4: 9 C-NOT gates are used for the complete four-qubit preparation circuit.The total is 1+2+3+3 gates across the four phases.
  • Phase 4: A circuit depth of 5 is achieved, with phase 2 taking one step and phases 3 and 4 running in parallel over three steps.This depth is optimal for a 9-C-NOT-gate circuit under the stated parallelism constraint.

FIVE QUBITS

For five qubits, the scheme factorizes the Hilbert space into two- and three-qubit parts and retains the first three phases while using a three-qubit unitary in phase four. It requires 26 C-NOT gates with depth 22.

  • FIVE QUBITS: The five-qubit Hilbert space is factorized into two- and three-qubit parts for Schmidt-based preparation.The Schmidt decomposition has at most four terms, with the |φ⟩i states now being three-qubit states.
  • FIVE QUBITS: A three-qubit unitary is performed in phase four, requiring no more than 20 C-NOT gates.The unitary is not completely defined because the third qubit initially occupies |0⟩, so further reduction may be possible.
  • FIVE QUBITS: 26 C-NOT gates are required for five-qubit preparation, matching the result of Ref..The count is 1+2+3+20 gates, while the lower limit is 13 C-NOT gates.
  • FIVE QUBITS: A depth of 22 computation steps is achieved, below the lowest known depth of 26 but above the theoretical lower bound of 7.Phases three and four run in parallel for 20 steps after one step each for phases one and two.

GENERAL CASE

The general construction factorizes the n-qubit Hilbert space into two parts, prepares generalized Schmidt coefficients on one part, entangles the parts with C-NOT gates, and finishes with local unitaries.

  • GENERAL CASE: For even n, the Hilbert space is factorized into two equal-dimensional parts; for odd n, the parts differ in size by one qubit.The first part contains n/2 qubits for even n and (n−1)/2 qubits for odd n.
  • GENERAL CASE: The first qubit subset is prepared with amplitudes determined by generalized Schmidt coefficients.These amplitudes are specified in the computational basis.
  • GENERAL CASE: C-NOT gates are applied between the two subsets, followed by separate unitary operations on each subset.The construction treats even and odd numbers of qubits separately.

Even number of qubits

For even n=2k, the scheme prepares an arbitrary state by splitting the qubits into two k-qubit halves, creating Schmidt coefficients, copying indices, and applying two k-qubit unitaries. It lowers the leading-order C-NOT bound and gives a reduced circuit depth.

  • Even number of qubits: n=2k qubits are divided into two k-qubit parts for a Schmidt-decomposition-based preparation scheme.The initial state is |0⟩⊗2k, with normalized k-qubit states and complex Schmidt coefficients.
  • Even number of qubits: The first phase prepares the Schmidt coefficients on the first k qubits using 2^k − k − 1 C-NOT gates.The amplitudes are prepared in the computational basis, with basis indices represented by binary strings.
  • Even number of qubits: The second phase uses k C-NOT gates from qubit j to qubit j+k, producing the desired Schmidt form.The remaining phases apply k-qubit unitary operations separately to the two halves.
  • Even number of qubits: 23/24 2^n is the leading-order upper bound for even n, obtained by combining the four preparation phases with universal k-qubit decompositions.The first phase contributes only order 2^(n/2), while phases three and four contribute order 2^n.
  • Even number of qubits: This is the new lowest number of C-NOT gates needed for a universal circuit preparing an arbitrary state.
  • Even number of qubits: The circuit depth is 23/48 2^n in the leading order, below the previous 2^n result but above the theoretical 2^n limit as transcribed.The cited passage reports these bounds for the depth of the construction.

Odd number of qubits

For odd n=2k+1, the first three phases remain unchanged, while the fourth phase acts on k+1 qubits. The resulting leading-order C-NOT and depth bounds are weaker than the even-qubit bound but remain below the best known result.

  • Odd number of qubits: n=2k+1 leaves the first three phases unchanged and applies the fourth phase to k+1 qubits.
  • Odd number of qubits: 96 2^n is the resulting leading-order C-NOT upper bound for odd numbers of qubits.The passage states that this bound is weaker than the even-qubit result.
  • Odd number of qubits: 192 2^n is smaller than the best known result for the circuit depth.

CONCLUSIONS

The scheme prepares arbitrary n-qubit states with an explicit C-NOT and one-qubit-rotation circuit, reducing both gate-count and computational depth. It also supports transformations between arbitrary given states and can benefit directly from improved arbitrary-gate decompositions.

  • The scheme prepares arbitrary n-qubit states using a gate library of C-NOT gates and one-qubit rotations.
  • Four-qubit preparation requires 9 C-NOT gates instead of 11 previously known.
  • Parallel execution of the final two phases yields roughly half the computational steps of previous results.
  • The results can help design and build small-scale quantum circuits using present technologies.
  • The procedure's efficiency follows arbitrary-gate decomposition results, so improved decompositions directly lower its bounds.
  • Reversing preparation for |ψ⟩ and then preparing |φ⟩ transforms any given state into any other, with less-than-doubled depth through parallel phases.
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