DatabaseSearch.qs

// Copyright (c) Microsoft Corporation.
// Licensed under the MIT License.
namespace Microsoft.Quantum.Samples.DatabaseSearch {
    open Microsoft.Quantum.Intrinsic;
    open Microsoft.Quantum.Convert;
    open Microsoft.Quantum.Math;
    open Microsoft.Quantum.Canon;
    open Microsoft.Quantum.Arrays;
    open Microsoft.Quantum.Measurement;
    open Microsoft.Quantum.Diagnostics;
    open Microsoft.Quantum.Oracles;
    open Microsoft.Quantum.AmplitudeAmplification;

    //////////////////////////////////////////////////////////////////////////
    // Introduction //////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////////

    // This sample will walk through several examples of searching a database
    // of N elements for a particular marked item using just O(1/√N) queries
    // to the database. In particular, we will follow Grover's algorithm, as
    // described in the standard library guide.

    // We will model the database by an oracle D that acts to map indices
    // to a flag indicating whether a given index is marked. In particular,
    // let |z〉 be a single-qubit computational basis state (that is, either
    // |0〉 or |1〉, and let |k〉 be a state representing an index k ∈ {0, 1,
    // …, N }. Then

    //     D |z〉 |k〉 = |z ⊕ xₖ〉 |k〉,

    // where x = x₀ x₁ … x_{N - 1} is a binary string such that xₖ is 1
    // if and only if the kth item is marked, and where ⊕ is the classical
    // exclusive OR gate. Note that given this definition, we know how D
    // transforms arbitrary states by linearity -- given an input state
    // that is a linear combination of orthogonal states |z〉|k〉 summed over the
    // z and k indices, D acts on each state independently.

    // First, we work out an example of how to construct and apply D without
    // using the canon. We then implement all the steps of Grover search
    // manually using this database oracle. Second, we show the amplitude
    // amplification libraries provided with the canon can make this task
    // significantly easier.

    //////////////////////////////////////////////////////////////////////////
    // Database Search with Manual Oracle Definitions ////////////////////////
    //////////////////////////////////////////////////////////////////////////

    // For the first example, we start by hard coding an oracle D
    // that always marks only the item k = N - 1 for N = 2^n and for
    // n a positive integer. Note that n is the number of qubits needed to
    // encode the database element index k.

    /// # Summary
    /// Given a qubit to use to store a mark bit and a register corresponding
    /// to a database, marks the first qubit conditioned on the register
    /// state being the all-ones state |11…1〉.
    ///
    /// # Input
    /// ## markedQubit
    /// A qubit to be targeted by an `X` operation controlled on whether
    /// the state of `databaseRegister` corresponds to a market item.
    /// ## databaseRegister
    /// A register representing the target of a query to the database.
    ///
    /// # Remarks
    /// Implements the operation
    ///
    ///     |z〉 |k〉 ↦ |z ⊕ f(k)〉 |k〉,
    ///
    /// where f(k) = 1 if and only if k = 2^(Length(databaseRegister)) - 1 and
    /// 0 otherwise.
    operation ApplyDatabaseOracle(markedQubit : Qubit, databaseRegister : Qubit[]) : Unit is Adj + Ctl {
        // The Controlled functor applies its operation conditioned on the
        // first input being in the |11…1〉 state, which is precisely
        // what we need for this example.
        Controlled X(databaseRegister, markedQubit);
    }


    // Grover's algorithm for quantum database searching requires that we
    // prepare the state given by the uniform superposition over all
    // computational basis states,

    //     |u〉 = Σₖ |k〉 = H^{⊗ n} |00…0〉,

    // where we have labeled n-qubit states by the integers formed by
    // interpreting their computational basis labels as big-endian
    // representations. For example, |2〉 in this notation is |10〉 in the
    // computational basis of two qubits.

    // Resolving this convention, then,

    //     |u〉 = |++…+〉.

    // This state is easy to implement given the input state |00…0〉, and we
    // call the oracle that does so U.

    /// # Summary
    /// Given a register of qubits initially in the state |00…0〉, prepares
    /// a uniform superposition over all computational basis states.
    ///
    /// # Input
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    operation ApplyUniformSuperpositionOracle(databaseRegister : Qubit[]) : Unit is Adj + Ctl {
        ApplyToEachCA(H, databaseRegister);
    }

    // We will define the state preparation oracle as a black-box unitary that
    // creates a uniform superposition of states using
    // `UniformSuperpositionOracle` U, then marks the target state using the
    // `DatabaseOracle` D. When acting on the input state |00…0〉, this prepares
    // the start state

    // |s〉 = D|0〉|u〉 = DU|0〉|0〉 = |1〉|N-1〉/√N + |0〉(|0〉+|1〉+...+|N-2〉)/√N.

    // Let us call DU the state preparation oracle. Note that if we were to
    // measure the marked qubit, we would obtain |1〉 and hence the index |N-1〉
    // with probability 1/N, which coincides with the classical random search
    // algorithm.

    // It is helpful to think of 1/√N = sin(θ) as the sine of an angle θ. Thus
    // the start state |s〉 = sin(θ) |1〉|N-1〉 + cos(θ) |0〉(|0〉+|1〉+...+|N-2〉)
    // is a unit vector in a two-dimensional subspace spanned by the
    // orthogonal states |1〉|N-1〉, and |0〉(|0〉+|1〉+...+|N-2〉).

    /// # Summary
    /// Given a register of qubits initially in the state |00…0〉, prepares
    /// the start state |1〉|N-1〉/√N + |0〉(|0〉+|1〉+...+|N-2〉)/√N.
    ///
    /// # Input
    /// ## markedQubit
    /// Qubit that indicates whether database element is marked.
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    operation ApplyStatePreparationOracle (markedQubit : Qubit, databaseRegister : Qubit[]) : Unit is Adj + Ctl {
        ApplyUniformSuperpositionOracle(databaseRegister);
        ApplyDatabaseOracle(markedQubit, databaseRegister);
    }


    // Grover's algorithm requires reflections about the marked state and the
    // start state. A reflection R is a unitary operator with eigenvalues ± 1,
    // and reflection about an arbitrary state |ψ〉 may be defined as

    // R = 1 - 2 |ψ〉〈ψ|.

    // Thus R|ψ〉 = -|ψ〉 applies a -1 phase, and R(|ψ〉) on any other state applies a
    // +1 phase. We now implement these reflections.

    /// # Summary
    /// Performs a reflection about the marked state.
    ///
    /// # Input
    /// ## markedQubit
    /// Qubit that indicated whether database element is marked.
    operation ReflectAboutMarkedState(markedQubit : Qubit) : Unit {
        // Marked elements always have the marked qubit in the state |1〉.
        R1(PI(), markedQubit);
    }

    /// # Summary
    /// Performs a reflection about the |00…0〉 state.
    ///
    /// # Input
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    operation ReflectAboutZero(databaseRegister : Qubit[]) : Unit {
        within {
            ApplyToEachCA(X, databaseRegister);
        } apply {
            Controlled Z(Rest(databaseRegister), Head(databaseRegister));
        }
    }


    /// # Summary
    /// Performs a reflection around the initial state, given by DU|0〉|0〉.
    ///
    /// # Input
    /// ## markedQubit
    /// Qubit that indicated whether database element is marked.
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    operation ReflectAboutInitialState(markedQubit : Qubit, databaseRegister : Qubit[]) : Unit {
        within {
            Adjoint ApplyStatePreparationOracle(markedQubit, databaseRegister);
        } apply {
            ReflectAboutZero([markedQubit] + databaseRegister);
        }
    }


    // We may then search our database for the marked elements by performing
    // on the start state a sequence of alternating reflections about the
    // marked state and the start state. The product RS · RM is known as the
    // Grover iterator, and each application of it rotates |s〉 in the two-
    // dimensional subspace by angle 2θ. Thus M application of it creates the
    // state

    // (RS · RM)^M |s〉 = sin((2M+1)θ) |1〉|N-1〉
    //                  + cos((2M+1)θ) |0〉(|0〉+|1〉+...+|N-2〉)

    // Observe that if we choose M = O(1/√N), we can obtain an O(1)
    // probability of obtaining the marked state |1〉. This is the Quantum
    // speedup!

    /// # Summary
    /// Prepares the start state and boosts the amplitude of the marked
    /// subspace by a sequence of reflections about the marked state and the
    /// start state.
    ///
    /// # Input
    /// ## nIterations
    /// Number of applications of the Grover iterate (RS · RM).
    /// ## markedQubit
    /// Qubit that indicated whether database element is marked.
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    operation SearchForMarkedState(nIterations : Int, markedQubit : Qubit, databaseRegister : Qubit[]) : Unit {
        ApplyStatePreparationOracle(markedQubit, databaseRegister);

        // Loop over Grover iterates.
        for (idx in 0 .. nIterations - 1) {
            ReflectAboutMarkedState(markedQubit);
            ReflectAboutInitialState(markedQubit, databaseRegister);
        }
    }

    // Let us now create an operation that allocates qubits for Grover's
    // algorithm, implements the `QuantumSearch`, measures the marked qubit
    // the database register, and returns the measurement results.

    /// # Summary
    /// Performs quantum search for the marked element and returns an index
    /// to the found element in binary format. Finds the marked element with
    /// probability sin²((2*nIterations+1) sin⁻¹(1/√N)).
    ///
    /// # Input
    /// ## nIterations
    /// Number of applications of the Grover iterate (RS · RM).
    /// ## nDatabaseQubits
    /// Number of qubits in the database register.
    ///
    /// # Output
    /// Measurement outcome of marked Qubit and measurement outcomes of
    /// the database register.
    operation ApplyQuantumSearch(nIterations : Int, nDatabaseQubits : Int) : (Result, Result[]) {
        // Allocate nDatabaseQubits + 1 qubits. These are all in the |0〉
        // state.
        using ((markedQubit, databaseRegister) = (Qubit(), Qubit[nDatabaseQubits])) {
            // Implement the quantum search algorithm.
            SearchForMarkedState(nIterations, markedQubit, databaseRegister);

            // Measure the marked qubit. On success, this should be One.
            let resultSuccess = MResetZ(markedQubit);

            // Measure the state of the database register post-selected on
            // the state of the marked qubit.
            let resultElement = ForEach(MResetZ, databaseRegister);

            // Returns the measurement results of the algorithm.
            return (resultSuccess, resultElement);
        }
    }


    // Here we test whether our hard coded-oracle is marking the right
    // fraction of bits

    /// # Summary
    /// Checks whether state preparation marks the right fraction of elements
    /// against theoretical predictions.
    operation StatePreparationOracleTest() : Unit {
        for (nDatabaseQubits in 0..5) {
            using ((markedQubit, databaseRegister) = (Qubit(), Qubit[nDatabaseQubits])) {
                ApplyStatePreparationOracle(markedQubit, databaseRegister);

                // This is the success probability as predicted by theory.
                // Note that this is computed only to verify that we have
                // implemented Grover's algorithm correctly in the
                // `AssertProb` below.
                let successAmplitude = 1.0 / Sqrt(IntAsDouble(2 ^ nDatabaseQubits));
                let successProbability = successAmplitude * successAmplitude;
                AssertMeasurementProbability([PauliZ], [markedQubit], One, successProbability, "Error: Success probability does not match theory", 1E-10);

                // This operation automatically resets all qubits to |0〉
                // for safe deallocation.
                Reset(markedQubit);
                ResetAll(databaseRegister);
            }
        }
    }


    // Here we perform quantum search using a varying number of iterations on
    // a database of varying size. Whenever the flag qubit indicates
    // success, we check that the index of the marked element matches our
    // expectations.

    /// # Summary
    /// Performs quantum search for the marked element and checks whether
    /// the success probability matches theoretical predictions. Then checks
    /// whether the correct index is found, post-selected on success.
    operation GroverHardCodedTest () : Unit {
        for (nDatabaseQubits in 0..4) {
            for (nIterations in 0..5) {
                using ((markedQubit, databaseRegister) = (Qubit(), Qubit[nDatabaseQubits])) {
                    SearchForMarkedState(nIterations, markedQubit, databaseRegister);
                    let dimension = IntAsDouble(2 ^ nDatabaseQubits);
                    let successAmplitude = Sin(
                        IntAsDouble(2 * nIterations + 1) *
                        ArcSin(1.0 / Sqrt(dimension))
                    );
                    let successProbability = PowD(successAmplitude, 2.0);
                    AssertMeasurementProbability([PauliZ], [markedQubit], One, successProbability, "Error: Success probability does not match theory", 1E-10);

                    // If this result is One, we have found the marked
                    // element.
                    let isMarked = MResetZ(markedQubit) == One;

                    if (isMarked) {
                        let results = ForEach(MResetZ, databaseRegister);

                        // Post-selected on success, verify that 
                        // database qubits are all |1〉.
                        for (result in results) {
                            EqualityFactR(result, One, "Found state should be 1..1 string.");
                        }
                    }
                }
            }
        }
    }


    //////////////////////////////////////////////////////////////////////////
    // Database Search with the Canon ////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////////

    // Our second example makes full use of the amplitude amplification
    // library and other supporting libraries to implement Grover's algorithm
    // more easily. We also consider a more general instance of the database
    // oracle that allows us to mark multiple elements.

    // The amplitude amplification library has a function called
    // ` StandardAmplitudeAmplification ` that automates many details of Grover search. Its
    // arguments have signature (Int, StateOracle, Int), where the first
    // parameter is the number of Grover iterates applied, the second
    // parameter is a unitary of type `StateOracle` and the third parameter
    // is an index to the MarkedQubit.

    // The state oracle is precisely the `StatePreparationOracle` operation we
    // defined above with one major difference -- Its arguments have signature
    // the signature (Int, Qubit[]). Rather than directly passing the marked
    // qubit to the operation, we instead pass an integer than indexes The
    // location of the marked qubit in the qubit array, which now encompasses
    // all qubits.

    // Our goal is thus to construct this `StateOracle` oracle type and pass it
    // to the ` StandardAmplitudeAmplification ` function. ` StandardAmplitudeAmplification ` acting on freshly
    // allocated qubits then automatically prepares a quantum state where the
    // marked subspace has been amplified.

    /// # Summary
    /// Database oracle `D` constructed from classical database.
    ///
    /// # Input
    /// ## markedElements
    /// Indices to marked elements in database.
    /// ## markedQubit
    /// Qubit that indicated whether database element is marked.
    /// ## databaseRegister
    /// A register of n qubits initially in the |00…0〉 state.
    ///
    /// # Remarks
    /// This implements the oracle D |z〉 |k〉 = |z ⊕ xₖ〉 |k〉 used in the Grover
    /// search algorithm. Given a database with N = 2^n elements, n is the
    /// size of the database qubit register. Let x = x₀x₁...x_{N-1} be a
    /// binary string of N elements. Then xₖ is 1 if k is in "markedElements"
    /// and 0 otherwise.
    operation ApplyDatabaseOracleFromInts(markedElements : Int[], markedQubit : Qubit, databaseRegister : Qubit[])
    : Unit
    is Adj + Ctl {
        for (markedElement in markedElements) {
            ControlledOnInt(markedElement, X)(databaseRegister, markedQubit);
        }
    }


    // The `StateOracle` described above is now constructed from partial
    // application of `GroverStatePrepOracle`. Note that we now index the
    // marked qubit with an integer.

    /// # Summary
    /// Preparation of start state from database oracle and oracle `U` that
    /// creates a uniform superposition over database indices.
    ///
    /// # Input
    /// ## markedElements
    /// Indices to marked elements in database.
    /// ## idxMarkedQubit
    /// Index to `MarkedQubit`.
    /// ## startQubits
    /// The collection of the n+1 qubits `MarkedQubit` and `databaseRegister`
    /// initially in the |00…0〉 state.
    ///
    /// # Remarks
    /// This implements an oracle `DU` that prepares the start state
    /// DU|0〉|0〉 = √(M/N)|1〉|marked〉 + √(1-(M/N)^2)|0〉|unmarked〉 where
    /// `M` is the length of `markedElements`, and
    /// `N` is 2^n, where `n` is the number of database qubits.
    operation _GroverStatePrepOracle(markedElements : Int[], idxMarkedQubit : Int, startQubits : Qubit[])
    : Unit
    is Adj + Ctl {
        let flagQubit = startQubits[idxMarkedQubit];
        let databaseRegister = Exclude([idxMarkedQubit], startQubits);

        // Apply oracle `U`
        ApplyToEachCA(H, databaseRegister);

        // Apply oracle `D`
        ApplyDatabaseOracleFromInts(markedElements, flagQubit, databaseRegister);
    }

    /// # Summary
    /// `StateOracle` type for the preparation of a start state that has a
    /// marked qubit entangled with some desired state in the database
    /// register.
    ///
    /// # Input
    /// ## markedElements
    /// Indices to marked elements in database.
    ///
    /// # Output
    /// A `StateOracle` type with signature
    /// ((Int, Qubit[]) => () is Adj + Ctl).
    function GroverStatePrepOracle(markedElements : Int[]) : StateOracle {
        return StateOracle(_GroverStatePrepOracle(markedElements, _, _));
    }


    // The library function ` StandardAmplitudeAmplification ` then returns a unitary that
    // implements all steps of Grover's algorithm.

    /// # Summary
    /// Grover's search algorithm using library functions.
    ///
    /// # Input
    /// ## markedElements
    /// Indices to marked elements in database.
    /// ## nIterations
    /// Number of iterations of the Grover iteration to apply.
    /// ## idxMarkedQubit
    /// Index to `MarkedQubit`.
    ///
    /// # Output
    /// Unitary implementing Grover's search algorithm.
    ///
    /// # Remarks
    /// On input |0〉|0〉, this prepares the state |1〉|marked〉 with amplitude
    /// Sin((2*nIterations + 1) ArcSin(Sqrt(M/N))).
    function GroverSearch(markedElements : Int[], nIterations : Int, idxMarkedQubit : Int)
    : (Qubit[] => Unit is Adj + Ctl) {
        return StandardAmplitudeAmplification(nIterations, GroverStatePrepOracle(markedElements), idxMarkedQubit);
    }

    // Let us now allocate qubits and run GroverSearch.

    /// # Summary
    /// Performs quantum search for the marked elements and returns an index
    /// to the found element in integer format.
    ///
    /// # Input
    /// ## markedElements
    /// Indices to marked elements in database.
    /// ## nIterations
    /// Number of applications of the Grover iterate (RS · RM).
    /// ## nDatabaseQubits
    /// Number of qubits in the database register.
    ///
    /// # Output
    /// Measurement outcome of marked Qubit and measurement outcomes of
    /// the database register converted to an integer.
    operation ApplyGroverSearch(markedElements : Int[], nIterations : Int, nDatabaseQubits : Int) : (Result, Int) {
        // Allocate nDatabaseQubits + 1 qubits. These are all in the |0〉
        // state.
        using ((markedQubit, databaseRegister) = (Qubit(), Qubit[nDatabaseQubits])) {
            // Implement the quantum search algorithm.
            GroverSearch(markedElements, nIterations, 0)([markedQubit] + databaseRegister);

            // Measure the marked qubit. On success, this should be One.
            let resultSuccess = MResetZ(markedQubit);

            // Measure the state of the database register post-selected on
            // the state of the marked qubit and return the measurement results
            // of the algorithm.
            return (resultSuccess, ResultArrayAsInt(ForEach(MResetZ, databaseRegister)));
        }
    }

}