Let me precise the answer form Annadurai with an example:

field_x = fields.Integer('Calculated Value', compute="_compute_field" search="_search_field")

@api.multi
def _search_field(self, operator, value):
    field_id = self.search([]).filtered(lambda x : x.field_x == value )
    return [('id', operator, [x.id for x in field_id] if field_id else False )]

I had same problem in having a computed field in filter

The only way is to define a non computed field from the computed one and put it in view. then it will appear in filter

Example:

payment_id = fields.Integer(compute='compute_lastpayment')

payment_state = fields.Char(string='Last Payment State')

    @api.multi

    def compute_lastpayment(self):

                for order in self:

                                if order.payment_ids:

                                                payment = order.payment_ids[0]

                                                order.write({'payment_state': payment.state})

<record model="ir.ui.view" id="purchase_order_tree_inherit">

        <field name="name">purchase.order.tree.inherit</field>

        <field name="model">purchase.order</field>

        <field name="inherit_id" ref="purchase.purchase_order_tree"/>

        <field name="arch" type="xml">

                 <field name="payment_id" invisible="1"/>

                <field name="payment_state"/>

            </field>

        </field>

    </record>

Only if you store the field.

See the example of the field END_DATE in the DEFAULT VALUES section of  http://www.odoo.com/documentation/10.0/howtos/backend.html#computed-fields-and-default-values

A compute field, by default, is not stored on database. You just need to add stored=True yo the field in Python file. That allows you to filter/group by computed field.

Tried add this attribute, but still did not show up in the filter dropdown:

approver_ids = fields.Many2many('res.users', compute="_get_approvers", string="Approvers", stored=True)

also tried

approver_ids = fields.Many2many('res.users', compute="_get_approvers", string="Approvers", stored="True")

However in the filter dropdown, there is no "Approvers" option